Lectures on Morse Theory II
H. Blaine Lawson Jr.
1
Lecture 1 - 1st February ’11
Let me begin by saying what I plan to do this term. It starts with René Thom. Consider a Morse function
f : X → R and let V be the vector field given by the gradient of f . Then for each critical point p ∈ Cr(f ), one has
local coordinates (u, v) with |u| < 1, |v| <, u(p) = 0 = v(p) and
f (u, v) = f (0, 0) − |u|2 + |v|2 , u = (u1 , . . . , uλ ), v = (v1 , . . . , vn−λ ).
v
Dn−λ
u
p
Dλ
Notice that grad(−|u|2 + |v|2 ) = 2(−u, v). V is called gradient like or good if
(i) V f > 0
(ii) V = (−u, v) in each coordinate system as above for each p ∈ Cr(f ).
We will also use ϕt to denote the flow of V .
Definition 1. (Stable/unstable manifolds)
For p ∈ Cr(f ) let
[
Sp ≡ {x ∈ X | lim ϕt (x) = p} =
ϕ−t (Dλ )
t→∞
t≥0
be the stable manifold of p.
The unstable manifold of p is defined by
Up ≡ {x ∈ X | lim ϕt (x) = p} =
t→−∞
[
t≥0
The example of a torus shows the stable and unstable manifolds.
a
b
c
d
The stable and unstable manifolds are contractible via the gradient flow.
2
ϕt (Dn−λ )
`
Remark 2. Since X = p∈Cr(f ) Sp , we may think of Sp as currents and the boundary of a stable manifold is again a
formal sum of stable manifolds. Then any gradient like vector field gives a cell decomposition of a manifold and one can
compute homology with integer coefficients.
The plan for this semester is as follows :
(1) Currents and forms - Federer lemma
(2) Finite volume flows
(3) Morse Theorem
(4) Equivariant Morse theory - Equivariant cohomology, Associated de Rham theory, Morse theory
(5) Novikov theory
(6) Equivariant Novikov theory
(7) Applications to symplectic geometry - Moment maps
(8) Morse-Bott theory
(9) Complex analytic case
Remark 3. (Novikov theory)
As an aside, Novikov sounds like a simple generalization of Morse theory although it is anything but. We study functions
f : X → S 1 . Pulling back the canonical form on S 1 we get an element H 1 (X; R). In fact, it is an integer cohomology
class. We ask that if f has non-degenerate critical points then is there a Morse theory? Generally, consider α ∈ E 1 (M ), a
smooth 1-form, dα = 0 with non-degenerate zeroes.
X
f
projection
S1
In the example above f ∗ (dθ) has no zeroes. Let me perturb the torus a bit and draw the picture of f from the universal
cover of the torus to the real line.
stable
f
Z
Let Λ ≡ Z[[t]][t−1 ] be the set of formal Laurent series. The starting point of Novikov theory is that there is a Morse
-Novikov complex.
3
Lift the critical points p1 , . . . , pN of f ∗ (dθ) to p̃1 , . . . , p̃N . Let
(
)
X
C =
nk ST k (p̃l ) .
ind(p̃l )=λ,k≥K
Although it is infinite dimensional, it is finitely generated free Λ-module.
Remark 4. (Symplectic geometry)
Let (X 2n , ω) be a symplectic manifold, i.e., ω ∈ E 2 (X) such that dω = 0 and ω n > 0. Let V be a vector field such that
ιV ω = dh. Then LV ω = 0 via Cartan’s formula. However, we may also consider vector fields V such that d(ιV ω) = 0.
This implies LV ω = 0 and is also where Novikov theory applies.
Remark 5. (Morse-Bott theory)
We look at the normal bundle N of a non-degenerate critical manifold.
N = N+ ⊕ N−
N ⊂ Cr(f )
N+
N−
The Hessian of f restricted to N is non-degenerate on each fibre. In particular, this machinery is going to give Geometric
Residue Theorems. It produces formulas explicitly at the level of forms and currents.
Remark 6. (Complex analytic case)
Consider a C∗ -action on a Kähler manifold. This implies a lot of structure. As an example, consider the following :
Theorem 7. Suppose X is a compact, Kähler manifold with a C∗ -action with isolated fixed points. Then all of H∗ (X; Z)
is generated by analytic subvarieties.
The theory shows that not only is this true but all of the cohomology are the same, the Chow group or K-theory. In fact,
all of the homology is of the type (p, p).
I have a few suggestions on what we could do later in the course :
(i) Cohen-Jones-Segal’s work on realizing a smooth manifold up to homeomorphism via Morse theory.
(ii) Infinite dimensional Morse theory on Hilbert manifolds.
(iii) Rips complexes (related to data analysis).
4
Lecture 2 - 3rd February ’11
This is a discussion of forms and currents and it’s well worth the revisit. Let X be a compact, oriented
n-dimensional manifold. Let E p (X) be the space of smooth p-forms on X. There is a chain complex
d
d
d
E 0 (X) −→ E 1 (X) −→ · · · −→ E n (X)
such that d2 = 0. There is a natural topology on this via uniform convergence of any finitely many derivatives on
compact subspaces. Let Ep0 (X) be the topological dual of E p (X). This is called the space of currents of dimension
p and degree n − p. The de Rham sequence has an adjoint sequence
∂
∂
∂
E00 (X) ←− E10 (X) ←− · · · ←− En0 (X).
By definition, (∂T )(ϕ) := T (dϕ) and it follows that ∂ 2 = 0.
Example 8. Let M ⊂ X be a p-dimensional, oriented C 1 -submanifold of finite volume1 . Define
Z
[M ](ϕ) :=
ϕ.
M
Check that
|[M ](ϕ)| ≤ vol(M ) sup kϕk
x∈X
whence [M ] is continuous. Suppose M is compact with boundary ∂M . Then
Z
Z
Stokes
(∂[M ])(ψ) := [M ](dψ) =
dψ =====
ψ = [∂M ](ψ),
M
∂M
i.e., ∂[M ] = [∂M ].
Now let M be a compact, oriented p-dimensional manifold equipped with a C 1 -map f : M → X. Then
f∗ [M ] ∈ Ep0 (X) given by
Z
(f∗ [M ])(ϕ) :=
f ∗ ϕ.
M
Let ∆ ⊂ R
p
p+1
be the standard p-simplex, i.e.,
∆p := {(x0 , . . . , xp ) | xj ≥ 0, x0 + · · · + xp = 1}.
For any map f : ∆p → X which is C 1 , we have [f ] ∈ Ep0 (X). In fact, one can do this with formal sums of such
p-simplices. Therefore, if one starts with a manifold with a triangulation, then its singular p-chains give rise to
currents of dimension p.
Example 9. Let ψ ∈ E n−p (X). Define [ψ] ∈ Ep0 (X) by
[ψ](ϕ) :=
Z
X
Notice that
(∂[ψ])(ϕ) = [ψ](dϕ) =
Z
X
ψ ∧ dϕ = (−1)n−p
Z
X
ψ ∧ ϕ.
d(ψ ∧ ϕ) − dψ ∧ ϕ = (−1)n−p+1
Z
X
dψ ∧ ϕ.
This implies that
∂[ψ] = (−1)n−p+1 [dψ].
Therefore, we have a natural embedding E n−p (X) ⊂ Ep0 (X) of complexes with ∂ = ±d. There is the classical result
of de Rham :
1 Put
a metric on X and get a volume form. Since X is compact, any two metrics are boundedly equivalent
5
Theorem 10. (de Rham)
H ∗ (E ∗ (X), d) ∼
= H ∗ (E 0∗ (X), d) ∼
= H ∗ (X; R).
Remark 11. If one knows the right proof of this, by this I mean sheaf-theoretic, then all of this can be proved in one fell
swoop. The sheaf of p-forms is a fine resolution of the constant sheaf R and that’s the usual de Rham’s theorem. For the
currential version, notice that any distribution that has all its derivatives zero is a constant distribution and it also gives
a fine resolution of R.
Note that if F : X → Y is smooth then F∗ : Ep0 (X) → Ep0 (Y ).
One should also make a mention of rectifiable currents.
Definition 12. A current T ∈ Ep0 (X) is rectifiable if there exists a countable family {Mk }∞
k=1 of mutually disjoint
oriented C 1 -submanifolds and positive integers {nk }∞
k=1 such that
∞
X
k=1
and T (ϕ) =
P∞
k=1
nk
R
Mk
nk vol(Mk ) < ∞
ϕ.
13. (Non-obvious) Any Lipschitz map F : B p (1) → X represents a rectifiable current via T ϕ :=
RExample
∗
F ϕ. This makes sense and is rectifiable essentially due to the fact that any Lipschitz map is almost everywhere
B
differentiable.
Definition 14. A current T ∈ Ep0 (X) is an integral current if T and ∂T are rectifiable.
Let Ip (X) be the p-dimensional integral currents on X equipped with
∂
∂
∂
In (X) −→ In−1 (X) −→ · · · −→ I0 (X).
Theorem 15. (Federer-Fleming)
H∗ (I∗ (X), ∂) ∼
= H∗ (X; Z).
Theorem 16. For K cpt ⊂ X and c > 0
{T ∈ Ip (X) | M (T ) + M (∂T ) < c, supp T ⊂ K}
is compact.
p
Suppose X is not compact. Let Ecpt
(X) be the space of p-forms with compact support on X and let Ep0 (X)
be its topological dual.
Definition 17. A linear map
p
T : Ecpt
(X) → R
is a current, i.e., continuous, if for each local coordinate system x = (x1 , . . . , xn ) on U open ⊂ Rn for X and each
domain Ω compactly contained in U and each k ∈ Z+ , there exists c > 0 such that
X
|T (ϕ)| ≤ C
sup kDα ϕk
|α|≤k
p
for any ϕ ∈ Ccpt
(Ω).
Definition 18. T has compact support if there exists K cpt ⊂ X such that
T (ϕ) = 0 ∀ ϕ with supp ϕ ⊂ X − K.
6
Let Ep0 (X)cpt denote the currents of dimension p with compact support.
Remark 19. If M p ⊂ X is an oriented C 1 -submanifold of locally finite2 volume then [M p ] ∈ Ep0 (X).
This is a basic theorem that I want to prove :
Theorem 20. Let M ⊂ X be a p-dimensional, oriented C 1 -submanifold of finite volume. Let O ⊂ X be an open set.
Let N ⊂ O be a closed submanifold which is connected, oriented and of dimension q. We also suppose that
supp {∂[M ]} ∩ O ⊂ N.
(i) If q < p − 1 then ∂[M ] = 0 in O.
(ii) If q = p − 1 then ∂[M ] = c[N ] in O for some constant c. In fact, this constant is an integer.
This is the result we’ll be using over and over again in Morse theory. Let me draw the pictures for the two cases
first :
(i)
(ii)
M
Mp
N p−1
N
This theorem is local in nature. We can assume that O is a coordinate bidisk, i.e.,
Proof
O = Dq × Dn−q = {(x, y) ∈ Rq × Rn−q | |x| < 1, |y| < 1}.
Furthermore, let N ∩ O = {(x, y) ∈ O | y = 0}. Define for 0 ≤ r ≤ 1 a map
Πr : O −→ O
given by
Πr (x, y) =
(x, (1 −
(x, 0) if |y| ≤ r
r
|y| )y) if r ≤ y < 1.
collapse
The first main lemma is the following :
Lemma 21. (i) Πr is 1-Lipschitz for all r.
(ii) Πr converge uniformly to Id as r → 0.
Rather than write a proof of this in the next few minutes, let me say where this is going.
2 For
any x ∈ X − M there exists a neighbourhood U open of x such that vol(U ∩ M ) < ∞.
7
Lemma 22.
lim (Πr )∗ [M ] = [M ].
r→0
If you apply a 1-Lipschitz map to a C 1 -submanifold, it gives a well defined current and this is not hard to prove.
Since ∂ is continuous, we have
Corollary 23.
lim ∂(Πr )∗ [M ] = [M ].
r→0
8
Lecture 3 - 8th February ’11
In the proof of the de Rham-Federer theorem, I had defined something called M . It’s known as the mass of
the current.
Definition 24. (Mass of a current)
For T ∈ Ep0 (X) define the mass M (T ) of the current T to be
M (T ) :=
sup T (ϕ),
M (ϕ)≤1
ϕ ∈ E p (X)
where
M (ϕ) := sup kϕx kmass
x∈X
p
and k · kmass is the norm in Λ
unit sphere.
Tx∗ X.
This is the supremum over a subset (consisting of decomposible vectors) of the
Notice that M (T ) < ∞ if and only if T extends to a continuous linear functional on all continuous p-forms.
Moreover,
|T (ϕ)| ≤ M (T ) sup kϕk.
X
→
−
→
−
If M (T ) < ∞ then there exists a Radon measure kT k; it is given by a section T of Λp T X with k T x k = 1 and
kT k a.e. such that
Z
→
−
T (ϕ) =
ϕ( T x )dkT k.
X
Let me get back to what I was saying the last time.
Theorem 25. If M p ⊂ X has finite volume and is oriented and if supp ∂M p ⊂ N q where N q is a q-dimensional
submanifold of X then
0 if q < p − 1
∂M p =
m[N ] if q = p − 1.
Let N ∩ O = {(x, y) ∈ O | y = 0}. Define for 0 ≤ r ≤ 1 a map
Πr : O −→ O
given by
Πr (x, y) =
(x, 0)
(x, (1 −
r
|y| )y)
collapse
The first main lemma is the following :
Lemma 26. (i) Πr is 1-Lipschitz for all r.
(ii) Πr converge uniformly to Id as r → 0.
9
if |y| ≤ r
if r ≤ y < 1.
Proof
Let z = (x, y), z 0 = (x0 , y 0 ) with |y|, |y 0 | ≥ r. Then
|Πr (z) − Πr (z 0 )|2
=
=
=
|Πr (z) − Πr (z 0 )|2 − |z − z 0 |2
=
=
=
2
r r |x − x0 |2 + 1 −
y − 1 − 0 y 0 |y|
|y |
2 2
r
r 0 r r
0 2
|x − x | + 1 −
y + 1 − 0 y − 2 1 −
1 − 0 y · y0
|y|
|y |
y
y
0
y
y
|x − x0 |2 + (|y| − r)2 + (|y 0 | − r)2 − 2(|y| − r)(|y 0 | − r)
· 0
|y| |y |
y
y0
(|y| − r)2 + (|y 0 | − r)2 − |y − y 0 |2 − 2(|y| − r)(|y 0 | − r)
· 0
|y| |y |
0
y
y
−2r(|y| + |y 0 |) + 2r2 − 2r(−|y| − |y 0 | + r)
·
|y| |y 0 |
y0 y
2r(r − |y| − |y 0 |) 1 −
· 0 ≤ 0.
|y| |y |
If |y| ≥ r, |y 0 | ≤ r then
2
r |Πr (z) − Πr (z )| = |x − x | + 1 −
y = |x − x0 |2 + (|y| − r)2 ≤ |z − z 0 |2
|y| 0
and if |y| ≤ r, |y 0 | ≤ r then
2
0 2
|Πr (z) − Πr (z 0 )|2 = |x − x0 |2 ≤ |z − z 0 |2 .
This is an elementary calculation but you wouldn’t see in anywhere in the literature. And it’s good to see it once.
Notice that
(0, y) if |y| ≤ r
z − Πr (z) =
r
0, |y|
y if |y| ≥ r.
Therefore, |z − Πr (z)| ≤ r for any z and we have uniform convergence.
Lemma 27. Let M ⊂ O − N be an oriented p-dimensional submanifold of finite volume. Then
lim (Πr )∗ [M ] = [M ].
r→0
Proof
Define
Hr : O × [0, 1] −→ O, Hr (z, t) := tz + (1 − t)Πr (z).
Set Mr := (Hr )∗ (M × [0, 1]). We claim that
lim vol(Mr ) = 0.
r→0
Notice that this proves that limr→0 Mr = 0 since
[Mr (ϕ) ≤ vol(Mr ) sup |ϕ| → 0.
O
q
n−q
We shall use Euclidean metric since all metrics are equivalent on O = D × D
. For an orthonormal basis
v1 , . . . , vp ∈ Tx M such that kv1 ∧ · · · ∧ vp k = 1 we have
Z
vol(Mr ) =
(Hr )∗ v1 ∧ · · · ∧ vp ∧ ∂t dvolM ×I
M ×[0,1]
Now
Dz Hr = t Id Rn + (1 − t)Dz Πr .
10
Since Πr is 1-Lipschitz, it is also almost everywhere differentiable on M by Rademacher’s theorem. This is clear
for any r which is a regular value of |y| restricted to M . This implies that
|(Hr )∗ v| ≤ |v| for all tangent vectors v to O.
Also notice that Dt Hr = Id − Πr whence
∂ r = 0, y ≤ r.
(Hr )∗
∂t
|y|
Therefore,
∂ ∂ (Hr )∗ v1 ∧ · · · ∧ vp ∧
≤ (Hr )∗ v1 · · · (Hr )∗ vp (Hr )∗ ≤ r
∂t
∂t
and vol(Mr ) converges to zero as r tends to zero. As a consequence ∂Mr → 0.
Assume further that supp(∂[M ]) ⊆ N in O. Then
∂Mr = ∂(Hr )∗ (M × [0, 1]) = [M ] − (Πr )∗ [M ].
Since ∂ is continuous on currents
lim ∂(Πr )∗ [M ] = ∂[M ].
r→0
We claim that
(πr )∗ (∂[M ]) =
0 if q < p − 1
m[Dq ] if q = p − 1.
We have supp ∂[M ] ⊂ N ⊂ Dq × Dn−q . Since Πr on Dq × Dn−q is the same as projection to Dq , we conclude
that
0
(Πr )∗ (∂[M ]) = pr∗ (∂[M ]) ∈ Ep−1
(Dq × {0}).
This is the key point - we get a current intrinsic to the submanifold. There is big difference between a current supported
on a submanifold and one that is intrinsic to it.
0
The first claim follows because there are no currents of dimension greater than q on Dq . Any T ∈ Ep−1
(Dp−1 )
0 0
p−1
with ∂T = 0 is the same as a function T ∈ (E ) (D
) with dT = 0, i.e., it is a constant. This is a basic result
in the theory of distributions which says that u ∈ (E 0 )0 (Rn ) with all its derivatives zero is a constant function.
For a proof look up Hörmander’s first book. It uses the idea of mollification, i.e., convolving with smooth bump
functions.
Operator Calculus
map
0
Let X m , Y n be compact, oriented smooth manifolds. Given T ∈ Em+r
(Y × X) define a continuous linear
T : E l (Y ) −→ (E 0 )l−r (X)
given by
T(α) := πX
Notice that (T ∧ ψ)(ϕ) := T (ψ ∧ ϕ).
∗
T
T ∧ πY∗ α .
πX
X
πY
Y
11
An alternative way to think about T is the following :
There are lots of examples of this.
∗
T(α)(β) = T πY∗ α ∧ πX
β .
Example 28. Let F : X → Y and let T be the current given by the graph of F . Then r = 0 and T(α) = F ∗ α.
12
Lecture 4 - 10th February ’11
Since there is a question on what locally finite volume manifolds look like let me give an example. Consider
the topologists sine curve
M := {(x, sin(1/x)) | x > 0} ⊂ R2 .
This does not have locally finite volume.
Operator Calculus
0
Let X m , Y n be compact, oriented manifolds. Then T ∈ Em+r
(Y × X) defines a continuous linear operator
l
0(l−r)
T : E (X) → E
(X). We’ve seen this last time. Let’s look at a few examples.
Example 29. Let F : X → Y be a smooth map and set T = [ΓF ] where
ΓF := {(F (x), x) | x ∈ X} ⊂ Y × X.
One checks that T(α) = F ∗ (α).
Example 30. Let Y = X and consider T = [∆], where ∆ is the diagonal in X × X. Then
T : E l (X) −→ E 0l (X)
is the identity map, i.e., it is the natural embedding of smooth forms into the space of generalized forms.
Example 31. Let Y = X and T = [U ] × [S], where U m−λ ⊂ X, S λ ⊂ X are manifolds of finite volume. Think
of T as an element of E 0m (X × X) and
Z R 0 if deg α 6= m − λ
T(α) :=
α [S] =
α if deg α = m − λ.
U
U
0
(Y × X). Then on E l (Y ) we have
Proposition 32. Let ∂T be the transformation associated to ∂T where T ∈ Em+r
∂T = (−1)l ∂ ◦ T + T ◦ d.
(1)
Proof
Let α ∈ E l (Y ), β ∈ E l−r+1 (X). Then
[∂T(α)](β)
=
=
=
=
∗
(∂T )(πy∗ α ∧ πX
β)
∗
T (d(πy∗ α ∧ πX
β))
∗
∗
T (πy∗ dα ∧ πX
β + (−1)l πy∗ α ∧ πX
dβ)
[T(dα)](β) + (−1)l [T(α)](dβ)
whence ∂T = (−1)l ∂ ◦ T + T ◦ d.
0
On Em+r−l
(X) = E 0l−r (X) set ∂ := (−1)l−r+1 d. Then
∂T = (−1)r+1 d ◦ T + T ◦ d
and for r odd, T is a chain homotopy between ∂T and the zero map.
Morse Theory
Let f : X → R be a Morse function on a compact, oriented manifold with a good gradient like flow ϕt : X → X.
Define
Γs := {(t, ϕt (x), x) ∈ R × X × X | t ∈ [0, s], x ∈ X}
and let Ts := pr∗ [Γs ] where pr : R × X × X → X × X.
13
X
Γs
∆
X
s
0
It follows that
∂Ts = [∆] − [Ps ].
Definition 33. The flow ϕt is a finite volume flow if
(i) for each critical point p ∈ crit(f ) Up and Sp have finite volume,
(ii) T := lims→∞ Ts has finite volume.
It’s good to note that real analytic submanifolds are always of finite volume.
Theorem 34. If ϕ is Morse-Smale then ϕ is finite volume.
0
We have T ∈ En+1
(X × X) with n = dim X and ∂T = [∆] − [P ]. This follows essentially from the fact
T = lims→∞ Ts exists and boundary is a continuous operator, i.e.,
∂T = lim ∂Ts = lim ([∆] − [Ps ])
s→∞
s→∞
exists. Recall that [Ps ](α) = ϕ∗s (α) which implies that
[P ](α) = lim ϕ∗s (α).
s→∞
In terms of operators3 , we have
d ◦ T + T ◦ d = I − P.
Theorem 35. We have the identification
P ∼
=
X
[Up ] × [Sp ].
p∈Cr(f )
In particular,
∂T = [∆] −
X
[Up ] × [Sp ].
p∈Cr(f )
For any α ∈ E l (X) one has
d ◦ T(α) + T(dα) = α −
X
p∈Cr(f )
(Z
Up
)
α [Sp ].
Remark 36. We’ll see later that I is chain homotopic to P where
P : E l (X) −→ ⊕p∈Cr(f ) R[Sp ] ≡ Sf .
The chain complex Sf is d-invariant, finite dimensional and is called the Morse complex.
Before we begin the proof we’ll need a definition.
Definition 37. Let x, y ∈ X. We say x < y if there exists a broken flow line
3 We
shall subsequently be writing P instead of [P ] interchangeably whenever the context of its usage is unambiguous.
14
x
y
from x to y.
We are excluding the relation x < x. Also note that if f is Morse-Smale then for p, q ∈ Cr(f ) with p < q we
conclude λp < λq .
Lemma 38. Let P ≡ lims→∞ Ps on X × X as before. Then
[
supp(P ) ⊂
p∈Cr(f )
ep := {x ∈ X | p < x}.
where U
15
ep × Sp
U
Lecture 5 - 15th February ’11
I want to elaborate on this fundamental lemma that I talked about last time.
Theorem 39. Let M ⊂ X be a p-dimensional, oriented C 1 -submanifold of finite volume. Let O ⊂ X be an open set.
Let N ⊂ O be a closed submanifold which is connected, oriented and of dimension q. We also suppose that
supp {∂[M ]} ∩ O ⊂ N.
(i) If q < p − 1 then ∂[M ] = 0 in O.
(ii) If q = p − 1 then ∂[M ] = c[N ] in O for some constant c. In fact, this constant is an integer.
Corollary 40. Let S = ∂T + R ∈ Ep−1 (O) where T = [M ] and M p is an oriented C 1 -submanifold of finite volume
in O − N and R = [L] with Lp−1 being an oriented C 1 -submanifold of finite volume in O − N . If
supp(∂S) ⊆ N q
and q < p − 1 then S = 0.
Proof We have this family of projections (in an ε-neighbourhood of N ) Πε : O → O. Using arguments similar
to proving
lim (Πε )∗ T = T
ε→0
one can prove that
lim (Πε )∗ R = R.
ε→0
The rest of the proof is easy enough.
Let f : X → R be a Morse function with a good gradient like flow ϕt . Let
Ts := {(ϕt (x), x) ∈ X × X | t ∈ [0, s]}.
We assume that
(i) vol(T∞ ) < ∞,
(ii) vol(Up ) < ∞ and vol(Sp ) < ∞ for p ∈ Cr(f ).
We had seen last time that the current Ps given by the graph of ϕs induces an operator
Ps : E ∗ (X) −→ E 0∗ (X).
Let P be the limit of Ps as s converges to infinity. We had also seen that
P(α) = lim ϕ∗s (α).
s→∞
Toe begin with, for each p ∈ Cr(f ) orient each Sp arbitrarily and orient each Up such that
or(Tp (Up ) × Tp (Sp )) = or(Tp X).
Then [Up × Sp ] = [Up ] × [Sp ] ∈ E 0n (X × X).
Theorem 41. We have the identification of currents
P =
X
[Up ] × [Sp ].
p∈Cr(f )
In particular,
∂T = [∆] −
X
[Up ] × [Sp ].
p∈Cr(f )
For any α ∈ E l (X) one has
d ◦ T(α) + T(dα) = α −
16
X
p∈Cr(f )
(Z
Up
)
α [Sp ].
Definition 42. Let x, y ∈ X. We say x < y if there exists a broken flow line from x to y.
x
y
By definition, we are excluding the relation x < x. Also note that if f is Morse-Smale then for p, q ∈ Cr(f ) with
p < q we conclude λp < λq .
Lemma 43. Let P ≡ lims→∞ Ps on X × X as before. Then
[
supp(P ) ⊂
p∈Cr(f )
ep := {x ∈ X | p < x}.
where U
Proof
ep × Sp
U
Clearly (y, x) ∈ supp(P ) only if there exists xk → x in X and tk → ∞ in R such that
yk := ϕtk (xk ) → y ∈ X.
Let L(xk , yk ) be the flow line from xk to yk . There exists c > 0 such that
length L(xk , yk ) ≤ c ∀ k.
Excise a neighbourhood U of p ∈ Cr(f ). For any given metric g on X, kV k ≥ c on X − U for some c. This isn’t
hard to see. Another thing that’s also easy is related to compactness : there exists a subsequence L(xk , yk ) such that
L(xk , yk ) converges to L(x, y), a broken flow line from x to y. This convergence is in any reasonable topology, for
example as 1-dimensional currents, and
∂L(x, y) = lim ∂L(xk , yk ) = lim {[yk ] − [xk ]} = [y] − [x].
k→∞
k→∞
Since tk → ∞, there must exist a critical point t ∈ int(L(x, y)). If not, then L(x, y) is an unbroken flow line from
ep .
x to y. Take p = limt→∞ ϕt (x). Then x ∈ Sp and y ∈ U
Sp
x
p
f
U
p
Up
q
Uq
y
Set
Σ :=
[
p<q
Uq × Sp , Σ0 := Σ ∪ {(p, p) | p ∈ Cr(f )}.
Then Σ0 is a finite disjoint union of submanifolds of dimension at most n − 1.
Notice that if p < q then λp < λq and the rest is clear. Let
T̊ ≡ (ϕt (x), x) | x ∈ X − Cr(f ), 0 < t < ∞ .
Note that T̊ is an embedded, oriented smooth submanifold of (X × X) − Σ0 .
(i) If (ϕt (x), x) = ϕt0 (x0 ), x0 ) then x = x0 and t = t0 since x 6∈ Cr(f ).
(ii) Define
F : (0, ∞) × X −→ X × X, F (t, x) = (ϕt (x), x).
17
If x is not critical then
F∗ (∂t ) = (V (ϕt (x)), 0) 6= 0.
Moreover, if v1 , . . . , vn is a basis of Tx X then
F∗ (vj ) = (∗, vj ), j = 1, . . . , n.
Therefore, F is not singular.
Our current T is just the integration over T̊.
Lemma 44. The closure T of T̊ in (X × X) − Σ0 is a proper, smooth, oriented submanifold with oriented boundary
[
Up × Sp .
∂T = ∆ −
p∈Cr(f )
Assume this lemma. Then T = [T̊ ] has
∂T = [∆] − P
in X × X. Then this lemma implies that
∂T = [∆] −
As a corollary, we conclude that
[
[Up × Sp ] in (X × X) − Σ0 .
p∈Cr(f )
X
supp P −
[Up × Sp ] ⊆ Σ0 .
p∈Cr(f )
Now apply the corollary to the Federer support theorem, i.e., set
X
∂T + R := ∂T + (P −
[Up × Sp ]).
p∈Cr(f )
Consequently,
P =
X
[Up × Sp ] in X × X.
p∈Cr(f )
18
Lecture 6 - 17th February ’11
I have two comments. Those who are taking notes should go back and change accordingly.
(1) Axioms for finite volume flow :
(i) vol(Up ) < ∞, vol(Sp ) < ∞ for any p ∈ Cr(f ).
(ii) vol(T̊∞ ) < ∞.
(iii) For any p, q ∈ Cr(f ), p < q implies λp < λq .
It’s a theorem that a Morse-Smale function gives a finite volume flow.
0
(2) On E l (X) ≡ En−l (X) ⊂ En−l
(X) let d = (−1)l+1 ∂. I will sometimes use d and sometimes use ∂. The
difference is just a sign and it usually doesn’t matter.
Let me quickly remind you what I did last time.
Lemma 45. Let P ≡ lims→∞ Ps on X × X as before. Then
[
supp(P ) ⊂
p∈Cr(f )
ep := {x ∈ X | p < x}.
where U
We set
Σ0 :=
[
q>p
ep × Sp
U
Uq × Sp ∪ {(p, p) | p ∈ Cr(f )}.
Lemma 46. The closure T of T̊ in (X × X) − Σ0 is a proper, smooth, oriented submanifold with oriented boundary
[
∂T = ∆ −
Up × Sp .
p∈Cr(f )
We applied these to T = [T̊ ] which has the current boundary ∂T = [∆] − P in X × X. By Lemma B,
X
∂T = [∆] −
[Up × Sp ]
p∈Cr(f )
in (X × X) − Σ0 . Consider
S ≡ ∂T − [∆] +
X
p∈Cr(f )
[Up × Sp ] = ∂T + R.
Then supp(S) ⊂ Σ0 and by the Federer support lemma
∂T + R = 0.
Proof of Lemma B We have seen that if (y, x) ∈ T − T̊ − Σ0 then either (y, x) = (x, x) ∈ ∆ or (y, x) ∈ Up × Sp
for some critical point p of f .
(1) Near ∆ ⊂ X × X,
T̊ : (ϕt (x), x) | 0 ≤ t < ε
clearly is a submanifold with boundary away from x ∈ Cr(f ).
(2) Given (y, x) ∈ Up × Sp , we apply the diffeomorphism ψt of X × X given by
ψt (z1 , z2 ) := (ϕ−t (z1 ), ϕt (z2 )).
Apply this for large t and we see that ψt (y, x) is contained in a small neighbourhood of (p, p). Since
ψt (T̊ ) = (T̊ ) ∀ t
19
it suffices to consider (y, x) in a small neighbourhood of (p, p). Choose the canonical coordinates (u, v, u, v) for a
neighbourhood U × U of (p, p) with
ϕt (u, v) = (e−t u, et v), ϕt (u, v) = (e−t u, et v).
Notice that T̊ in U × U is given by
{(e−t u, et v, u, v) | |u| ≤ 1, |v| ≤ 1, 0 ≤ t ≤ |v|}.
In other words, it is the set of points (su, s−1 v, u, v) where s = et ∈ (0, 1].
add picture
This completes the proof of Lemma B.
We have proved that
∂T = [∆] −
X
p∈Cr(f )
[Up × Sp ].
This equation in terms of operators looks like
d ◦ T + T ◦ d = I − P.
where T : E ∗ (X) ,→ E 0∗ (X) is the canonical inclusion and
(Z
X
P(α) :=
p∈Cr(f )
Up
)
α [Sp ].
Definition 47. Consider the finite dimensional linear subspace
M
Sf :=
R[Sp ] ⊂ E∗0 (X).
p∈Cr(f )
It is called the Morse complex associated with f .
Notice that P : E ∗ (X) → Sf and ∂ ◦ P = P ◦ ∂ implies that (Sf , ∂) is a subcomplex of (E∗0 , ∂). This means that
the boundary of every stable manifold (as a current) is a combination of stable manifolds.
Theorem 48. The surjective linear map
P̂ : E ∗ (X) −→ Sf
induces an isomorphism
∼
=
∗
HdeR
(X) −→ H ∗ (Sf ).
Proof
Since I is chain homotopic to P we have
I∗ = P∗ : H∗ (E∗ (X)) −→ H∗ (E∗0 (X)).
We have seen that I∗ is an isomorphism. The map P factors through P̂ : E∗ (X) → Sf and we conclude that P̂∗ is
injective. We only need to prove surjectivity, i.e., given s ∈ Sfk such that ds = 0 then s = P(gamma) for some
γ ∈ E n−k (X).
Observe that for any p, q ∈ Crλ (f ) of the same index
(i) S p ∩ ∂Uq = φ
(ii) S p ∩ U q 6= φ if p 6= q
(iii) S p ∩ U p = {p}.
Recall that if r < s then λr < λs . To see (i) observe that
[
[
∂Uq ⊆
Ur ⊆
Ur
r>q
λr >λq =λ
20
while
Sp ⊆
[
r≤p
Sr ⊆
[
Sr
λr ≤λp =λ
and these are clearly disjoint. The proofs for (ii) and (iii) are similar.
Let |S| be the support of S and pick an open neighbourhood N such that (by using (i))
N ∩ ∂Uq = φ ∀ λq = λ.
By a variant of deRham’s theorem,
0
H(Ecpt (N )) ∼
(N )).
= H(Ecpt
0(k−1)
This implies there exists a smooth current (form) γ where dγ = 0 and σ ∈ Ecpt
(N ) such that
γ − S = dσ.
The idea (which I’ll take up next time) is that Up is a closed submanifold of N and one can construct a smooth
Thom form supported in an ε-neighbourhood of Up such that it is closed and it converges to Up as a current as
ε → 0.
21
Lecture 7 - 22nd February ’11
We have seen last time that
d ◦ T + T ◦ d = I − P.
We also noticed that P factors through P̂ and we wanted to prove that P̂∗ is an isomorphism. We had seen that it is
injective and it remains to show that it is surjective, i.e., given s ∈ Sfk such that ∂s = 0 then s = P(γ) for some
γ ∈ E k (X) with dγ = 0. Observe that for any p, q ∈ Crλ (f ) of the same index
(i) S p ∩ ∂Uq = φ
(ii) S p ∩ U q 6= φ if p 6= q
(iii) S p ∩ U p = {p}.
Recall that if r < s then λr < λs . This follows from the axioms of finite volume flow. To see (i) observe that
[
[
∂Uq ⊆
Us ⊆
Us
s>q
while
Sp ⊆
[
r≤p
λs >λq =λ
Sr ⊆
[
Sr
λr ≤λp =λ
and these are clearly4 disjoint. Therefore, there exists a neighbourhood N of supp(S) such that
N ∩ ∂Uq = φ ∀ q ∈ Crk (f ).
N
P
U
S
0(k−1)
k
(N ) with dγ = 0 and σ ∈ Ecpt
By deRham’s theorem, there exists a smooth form γ ∈ Ecpt
(N ) such that
γ − s = dσ.
For each q ∈ Cr(f ) of index k, Uq ∩ N is a closed submanifold of dimension n − k in N .
There exists a family {τεq }ε>0 of smooth closed k-forms in N (called Thom forms) such that
ε→0
τε −→ Uq in E 0k (N ).
An easy proof of this is given by first realizing this for submanifolds in Rn via convolution. In general, embed
X ,→ Rn , convolve with a smooth form and then restrict.
lim (Sp , τεq ) = (Sp , Uq ) = Uq · Sp = δpq ∀ p ∈ Crk (f ).
ε→0
Write S =
P
rp [Sp ] and notice that
(γ − s, Uq ) = lim (γ − s, τεq ) = lim (dσ, τεq ) = lim (σ, dτεq ) = 0.
ε→0
ε→0
ε→0
4 It is immediate for Morse-Smale by transversality. The general case is actually easier just by assuming p > q and the intersection hypothesis
gives us q < p, a contradiction!
22
This implies that
Z
γ = (S, Uq ) =
Uq
whence s = P(γ).
Now I want to prove this over Z. Let
X
rp [Sp ], Up ,
SfZ := ⊕p∈Cr(f ) Z[Sp ]
be a lattice in Sf .
Theorem 49. SfZ is subcomplex of Sf . Moreover, the inclusion
SfZ ,→ I∗ (X)
into integral currents induces an isomorphism
∼
=
ι∗ : H∗ (SfZ ) −→ H∗ (X; Z).
Proof
We know that
X
∂[Sp ] =
λq =λp −1
npq [Sq ], npq ∈ R.
There are two ways to prove this.
(i) By the big theorem of Federer, R is a rectifiable current and M (∂R) < ∞ then ∂R is rectifiable and npq ∈ Z.
(ii) When ϕt is Morse-Smale, we calculate directly. We reverse the flow line and the boundary of a small disk in
Sp (which has a natural orientation) which gets carried diffeomorphically (at least locally) to Sq .
p
Sp
Uq
Sq
q
Define
nγ =
1
−1
It’s then easy to check that
npq =
if or(∂Sp ) = or(Sq )
if not.
X
nγ .
γ is a flow line from p to q
We have shown that this is a subcomplex. For the homology isomorphism there are two proofs.
(i) Extend the domain of T, P to include all C ∞ singular chains which are transversal to Uq for any q. Apply
23
d ◦ T + T ◦ d = I − P to integer chains.
(ii) Fix a smooth triangulation τ of X. Let C∗ (τ ) be the integer chain groups of τ . We deform this such that
every simplex is transversal to every Up , p ∈ Cr(f ). We do this by choosing V1 , . . . , VN of vector fields such that
these span Tx X at every x ∈ X. Let ϕ1,t , . . . , ϕN,t be the corresponding flows. Set
Φ(t1 ,...,tN ) : X −→ X, Φ := ϕ1,t1 ◦ · · · ◦ ϕ1,tN .
The transversality theorem for families imply that for almost all t := (t1 , . . . , tN )
Φt (∆) t Uq ∀ ∆ ∈ τ, q ∈ Cr(f ).
Deform τ to τe := Φt (τ ) by such a map. Then the operator P is well defined on C∗ (τ ) ⊂ E∗0 (X), i.e.,
X
P(∆k ) =
(∆k ∩ Up )[Sp ].
p∈Crk (f )
The chain homotopy formula shows that I∗ = P∗ on the subcomplex C∗ (τ ). But I∗ is already an isomorphism and
we’re done.
This has a number of nice corollaries.
Corollary 50. For any abelian group G we have the isomorphism
H∗ (SfZ ⊗ G) ∼
= H∗ (X; G).
We have the short exact sequence
0 −→ E0∗ (X) −→ E ∗ (X) −→ Sf −→ 0
P
where E0∗ (X) = {α | α · Up = 0 ∀ p ∈ Cr(f )} is of finite codimension. The long exact sequence in homology
implies that
H∗ (E0∗ (X)) = 0.
We’ll talk about integral homology next time via SfZ . We shall see that
∼
=
P : H(EZ∗ (X)) −→ H(SfZ ) ∼
= H(X; Z).
In particular, given s ∈ Sfk we can find α ∈ EZk with dα = 0 such that
α − s = d(T(α)).
Here α is smooth, s is a sum of submanifolds and T(α) is a spark, appearing also in the theory of differential
characters.
24
Lecture 8 - 24th February ’11
We will talk about Poincaré duality today. Let f : X → R be a Morse-Smale function on a smooth manifold
X and consider
SfZ := ⊕p∈Cr(f ) Z[Sp ], UfZ := ⊕p∈Cr(f ) Z[Up ].
These are chain complexes and we have
H(SfZ , ∂) ∼
= H(UfZ , ∂) ∼
= H(X; Z).
We oriented {Sp }p∈Cr(f ) arbitrarily. We oriented {Up }p∈Cr(f ) such that
Up · Sq = δpq ∀ p, q ∈ Crk (f ) for any k.
Notice that Up · Sq is the intersection product. By construction, {Up }p∈Cr(f ) is a basis for the cochain complex
UfZ = Hom(SfZ , Z). Let δ be the adjoint of ∂ on UfZ . This gives us
∗
Hn−∗
(UfZ ) ∼
= H ∗ (X; Z).
However, if we interchange roles, we know
H∗ (UfZ , ∂) ∼
= H∗ (X; Z).
We need to show that δ = ∂ up to a global sign which will imply
Hn−∗ (X; Z) ∼
= H ∗ (X; Z).
This holds over Z2 .
There is an interesting fact. Let
Ttot := {(ϕt (x), x) | x ∈ X, t ∈ R} = T − + T +
where T − := Ttot |t≤0 and T + := Ttot |t≥0 . Note that
X X ∂Ttot =
Up × Sp −
Sp × Up
p∈Cr(f )
gives an operator Ttot such that
e are given by
The operators P, P
p∈Cr(f )
e
d ◦ Ttot + Ttot ◦ d = P − P.
P(α)
=
X nZ
p∈Cr(f )
e
P(α)
=
Up
X nZ
p∈Cr(f )
Sp
o
α [Sp ]
o
α [Up ],
e projects on Sf (respectively Uf ). This shows that P∗ = P
e∗ on H∗ . This works over Z, Q
where P (respectively P)
or Z2 -coefficients as well.
Let α ∈ E k (X) such that dα = 0 and α has Z-residues. By definition,
X
(δUq ) · Sp := Uq · (∂Sp ) = Uq ·
npr Sr = npq .
r
25
Lemma 51. The following identity holds :
(∂Uq ) · Sp = (−1)n−λ+1 Uq · (∂Sp )
where dim Sp = λ and dim Uq = n − λ + 1.
If we write
∂Uq =
X
ñqs Us
s
then we need to show
npq = (−1)n−λ+1 ñqp .
There are two ways of proving this.
First proof
One just needs to just check some signs keeping in mind the following :
p
Sp
Uq
Sq
q
Notice that nγ = 1 if ∂Sp has orientation agreeing with Sq and equals −1 otherwise. Now
1
=
or(Tq X)
=
or(Uq ) ⊕ or(Sq )
=
or(Uq ) ⊕ nγ or(∂Sp )
or(Uq ) ⊕ or(∂Sp ).
nγ
=
1
=
or(Tp X)
=
or(Up ) ⊕ or(Sp )
In a similar manner,
=
ñγ
=
ñγ or(∂Uq ) ⊕ or(Sp )
or(∂Uq ) ⊕ or(Sp ).
26
γ
Sp
Uq
We have that
or(Sp )
=
or(Uq )
=
or(~γ ) ⊕ or(∂Sp )
or(~γ ) ⊕ or(∂Uq ).
Using this we see that
nγ
=
=
=
=
or(Uq ) ⊕ or(∂Sp )
(−1)n−λ+1 or(∂Uq ) ⊕ or(~γ ) ⊕ or(∂Sp )
(−1)n−λ+1 or(∂Uq ) ⊕ or(Sp )
(−1)n−λ+1 ñγ .
This finishes the proof.
Second Proof
5
We set
Uq (t)
:= ϕt (Uε (q))
Sp (t)
:= ϕ−t (Sε (p)).
Sp(t)
p
Uq (t)
Sp
Up
Uq
q
Sq
Notice that [Uq (t)] ∧ [Sp (t)] is a finite number of flow line segments (arcs) contained in the flow lines γ from q to
5 Assume
that the flow is Morse-Smale.
27
p.
npq
=
ñpq
=
0
=
=
=
Z
ZX
[Uq (t)] ∧ ∂[Sp (t)]
∂[Uq (t)] ∧ [Sp (t)]
ZX d [Uq (t)] ∧ [Sp (t)]
ZX
Z
d[Uq (t)] ∧ [Sp (t)] + (−1)λ−1
[Uq (t)] ∧ d[Sp (t)]
X
X
Z
Z
(−1)λ
∂[Uq (t)] ∧ [Sp (t)] + (−1)λ−1 (−1)n−λ+1
[Uq (t)] ∧ ∂[Sp (t)].
X
X
This gives us the required identity.
It should be observed that
lim ϕ∗t (α) =: P(α) ∈ Sf
t→∞
is a cycle class in H∗ (X; Z) with its dual cocycle given by
e
lim ϕ∗t (α) =: P(α)
∈ Uf .
t→−∞
Also note that Z-residues mean that evaluating α over Sp or Up results in integers.
Equivariant Cohomology
Let G be a compact Lie group and X a topological space homeomorphic to a countable CW complex.
Definition 52. X is a G-space if there is a continuous action of G on X, i.e., e · x = x and g1 · (g2 · x) = (g1 g2 ) · x
for any g1 , g2 ∈ G.
Definition 53. A G-map (between X and X 0 ) is a continuous map f : X → X 0 between G-spaces, i.e., f (g · x) =
g · f (x) for any g ∈ G, x ∈ X.
Such an f as above is also called a G-equivariant map.
∗
The idea is to devise a cohomology functor HG
(·) on this category of topological spaces. If G acts freely on
X then
∗
HG
(X) ∼
= H∗ (X/G).
A free action is the same as X → X/G is a principal G-bundle.
Example 54. The permutation group Σn acts on coordinates in Cn and therefore on S 2n−1 ⊂ Cn . Any finite
group G ⊂ Σn for some n thereby making S 2n−1 a G-space.
Definition 55. (The Borel construction)
Let E be a contractible G-space on which G acts freely and E → E/G is a principal G-bundle. Given any G-space
X, the diagonal action of G on X × E is free. We set
∗
HG
(X) := H ∗ (XG ), XG := (X × E)/G.
This construction desingularizes the action.
28
Lecture 9 - 1st March ’11
As we mentioned last time, let G be a compact Lie group. We also assume our spaces are in the category
of countable CW complexes. Let G act on X, i.e., there is a continuous map G × X → X satisfying the usual
properties.
Definition 56. X is a G-space if there is a continuous action of G on X, i.e., e · x = x and g1 · (g2 · x) = (g1 g2 ) · x
for any g1 , g2 ∈ G.
Definition 57. A G-map (between X and X 0 ) is a continuous map f : X → X 0 between G-spaces, i.e., f (g · x) =
g · f (x) for any g ∈ G, x ∈ X.
Such an f as above is also called a G-equivariant map.
∗
The idea is to devise a cohomology functor HG
(·) on this category. If G acts freely on X then
∗
HG
(X) ∼
= H∗ (X/G).
A free action is the same as X → X/G is a principal G-bundle.
Definition 58. (The Borel construction)
Let E be a contractible G-space on which G acts freely and E → E/G is a principal G-bundle. Given any G-space
X, the diagonal action of G on X × E is free. We desingularize the G-action by considering
∗
HG
(X) := H ∗ (XG ), XG := (X × E)/G.
One thing is true : suppose G acts freely and X × E → X induces a fibre bundle XG → X/G with fibre E. Since
E is contractible, π∗ (E) ≡ 0 and by the long exact sequence
∼
=
π∗ (XG ) −→ π∗ (X/G).
∗
(·)
Therefore, XG → X/G is a weak homotopy equivalence. However, we need to show that this definition of HG
is independent of choice of E and that E exists.
π
Proposition 59. Let Y −→ B be a fibre bundle over a countable CW complex with contractible fibre E. Then there
exists a global section σ : B → Y , i.e., π ◦ σ = IdB , and any two such cross-sections are homotopic.
Proof
Proceed inductively on the skeleta of B. Let Bk be the k-skeleton of B and write
Bk = Bk−1 ∪ϕ1 D1k ∪ϕ2 D2k ∪ · · ·
where ϕj : ∂Djk → Bk−1 . Give B the weak limit topology.6
Lemma 60. Given a continuous map Φ : Dk → B, the induced bundle Φ∗ (Y ) is trivial.
Proof Note that if Φ0 and Φ1 are homotopic maps then Φ∗0 (Y ) ∼
= Φ∗1 (Y ) as bundles. Then observe that any map
is homotopic to a constant map and we’re done.
Φ
We now apply this to D = Djk −→ B where Φ = ϕj on ∂D and Φ is a homeomorphism on to its image in D̊. We
have
∼
=
Φ∗j (Y ) −→ Dk × E.
Given the section σ over Bk−1 (bu induction), we pull back by Φj and have a section on ∂Dk . Since the fibre E is
contractible, we extend this section to Dk .
Now given two sections σ0 , σ1 , assume a homotopy from σ0 to σ1 is constructed on Bk−1 . Pull back σt back
to Φ∗j (Y ) ∼
= Dk × E. We have s : ∂([0, 1] × Dk ) → E and extend over [0, 1] × Dk as E is contractible.
There is another point in this picture.
6 Given a space B and compact Hausdorff spaces K ⊂ K ⊂ · · · such that B is the union of these compact spaces. Declare C ⊂ B to be
1
2
closed if and only if C ∩ Kj is closed for all j. Recall that a closed set K ⊂ B is compact if and only if K ⊂ Kj for some j.
29
π
Proposition 61. Suppose E → E/G is a principal G-bundle with E contractible. Let Y −→ Y /G be any principal
G-bundle. Then there exists a natural bijective correspondence between
n
o
π̃
cross-sections of the bundle (Y × E)/G −→ Y /G ←→ G-equivariant maps Φ : Y → E .
Proof A section σ : Y /G → (Y × E)/G assigns to x ∈ Y /G a unique orbit {(gy, ge) | g ∈ G} ⊂ Y × E such
that π(y) = x. This orbit is an equivariant map of π −1 (x) → E given by y 7→ e. Assemble these over Y /G and get
an equivariant map Y → E. The converse is easy.
Theorem 62. Let E → E/G be a principal G-bundle with E contractible. Let Y → Y /G be any principal G-bundle.
Then there exists a G-equivariant map Y → e which is unique up to G-equivariant homotopy.
Proof The first proposition gives a cross-section σ : (Y × E)/G → Y /G which is unique up to homotopy and
then apply the second proposition.
Corollary 63. Let E → E/G and E 0 → E 0 /G be principal G-bundle with E, E 0 contractible. Then E and E 0 are
G-equivariantly homotopy equivalent.
Proof It follows from the theorem that there exists G-equivariant maps Φ : E → E 0 , Φ0 : E 0 → E. By the
corollary, Φ0 ◦ Φ : E → E, Φ ◦ Φ0 : E 0 → E 0 are unique up to G-equivariant homotopy, i.e.,
Φ0 ◦ Φ ∼ IdE , Φ ◦ Φ0 ∼ IdE 0
via G-equivariant homotopy.
Theorem 64. (Classification of principal bundles)
Let E → E/G be a principal G-bundle with E contractible. Then for every principal G-bundle Y → Y /G there exists
a continuous map
ϕY : Y /G → E/G
unique up to homotopy such that ϕ∗Y (E) ∼
=Y.
Proof By the previous theorem, there exists a G-equivariant map, unique up to homotopy, Φ : Y → E. This
induces a map ϕY : Y /G → E/G. The map
Y → ϕ∗Y (E), y 7→ (πY (y), Φ(y))
is a G-equivariant homeomorphism.
30
Lecture 10 - 3rd March ’11
The up shot of what I proved last time were the following two basic conslusions :
(1) The principal G-bundle EG → BG := EG /G, where EG is contractible, is unique up to G-homotopy
equivalence.
(2) For any space X, there is a natural bijection between equivalence classes of principal G-bundles on X and
homotopy classes of maps from X to BG . The space BG is called the classifying space for principal G-bundles.
(3) For any ring Λ
∗
HG
(pt; Λ) := H ∗ (BG ; Λ)
is well defined.
(4) For any u ∈ H k (BG ; Λ) we have a universal characteristic class for principal G-bundles. Given P → X, a
principal G-bundle, let fP : X → BG be the classifying map. Then
u(P ) := fP∗ (u) ∈ H k (X; Λ).
Now back to the question of existence.
Theorem 65. Let G be a compact Lie group. Then there exists a principal G-bundle EG → BG with EG contractible.
Proof
Recall that every compact Lie group has a faithful unitary representation, i.e.,
ϕ : G ,→ U (n)
is an injective group homomorphism for some n. If E = EUn is a contractible principal Un -bundle then restricting
the action to G we have a principal G-bundle E → E/G. Therefore, it suffices to construct EUn → BUn .
Fix a large integer n ∈ Z+ and fix CN with the standard hermitian metric (·, ·). Let
Stn,N := {(e1 , . . . , en ) ∈ CN × · · · × CN | (e1 , ej ) = δij }
be the space of hermitian orthonormal n-frames in CN . The group Un acts freely, i.e., given g = ((gij )) ∈ Un
g · (e1 , . . . , en ) = (ge1 , . . . , gen ).
We also have the fibration
span
Stn,N −→ Stn,N /Un =: Gn,N .
This is a principal Un -bundle. It is the same as the fibration
UN /UN −n −→ UN /(Un × UN −n ).
Notice that the fibration
Stn,N −→ S 2N −1 , (e1 , . . . , en ) 7→ en
with fibre Stn−1,N −1 . The long exact sequence of homotopy groups gives us :
· · · −→ πk+1 S 2N −1 −→ πk Stn−1,N −1 −→ πk Stn,N −→ πk S 2N −1 −→ · · ·
This implies that
πk Stn−1,N −1 ∼
= πk Stn,N if k ≤ 2n − 3.
As a consequence,
πk Stn,N ≡ 0 if k ≤ 2(N − n).
We define Stn,∞ as the direct limit of Stn,N endowed with the weak topology. It is clear that this space has no
homotopy groups and by the Whitehead theorem, this space is contractible. Check that
Stn,∞ −→ Gn,∞ = lim Gn,N
N →∞
31
is a principal Un -bundle.
There is also the famous join construction due to Milnor. Let ∗ denote the join of two spaces. Set
E(n) = G
∗ · · · ∗ G}
| ∗ G {z
N times
and define EG = limN →∞ E(n). Check that there is a free G-action on EG .
Let’s get back to equivariant cohomology. Let X be a G-space. Then XG := (X × EG )/G is well defined
up to homotopy equivalence and so is H ∗ (X) := H ∗ (XG ). There are two maps
XG?
?? ρG

??


?

BG
X/G
π
Notice that ρG is a fibre bundle with fibre X and H ∗ (XG ) is a H ∗ (BG )-module via ρ∗G . If X → X/G is a principal
∗
∗
bundle then π is a homotopy equivalence and HG
(X) ∼
(·) is functor from the category of
= H ∗ (XG ). So HG
∗
G-spaces and G-equivariant maps to H (BG )-modules.
Example 66. If X = G then
∗
HG
(G) = H ∗ (pt) =
Λ
0
if k = 0
if not.
If X = pt then
∗
HG
(pt) = H ∗ (BG ; Λ).
Example 67. If G = Un then
HU∗ n (pt) = H ∗ (BUn ; Z) = Z[c, . . . , cn ]
where ck ∈ H 2k (BUn ; Z) is the kth universal Chern class.
Example 68. If G = On then
H ∗ (BOn ; Z2 ) ∼
= Z2 [w1 , . . . , wn ]
where wk ∈ H k (BOn ; Z2 ) is the universal Stiefel-Whitney class.
Example 69. If G = SO2n then
H ∗ (BSO2n ; R) ∼
= R[p1 , . . . , pn , χ]/(χ2 = pn )
where pi ’s are the Pontrjagin classes and χ is the Euler class.
Example 70. Let Γ ,→ Un be a finite subgroup, i.e., we have a finite dimensional representation of Γ. We have an
induced fibration
BΓ = EUn /Γ −→ EUn /Un = BUn
with fibre Un /Γ. We pull back the Chern classes c1 , . . . , cn to H ∗ (BΓ ) to get Chern classes for this representation.
∗
Example 71. Let K ⊂ G be a compact subgroup and consider the homogeneous space G/K. Then HG
(G/K) =
∗
H ((G/K × EG )/G). Notice that G × EG is a G × K-space with G acting diagonally and K acting on G on the
right. Since (G × EG )/G ∼ EG and it is contractible on which K acts freely, we conclude that
∗
∗
HG
(G/K) ∼
(pt).
= HK
32
Lecture 11 - 8rd March ’11
Last time I gave some examples of equivariant cohomology. Here’s another :
Example 72. Suppose G acts trivially on X. Then
H ∗ (X; R) = H ∗ ((X × EG )/G; R) = H ∗ (X × BG ; R) = H ∗ (X; R) ⊗ H ∗ (BG ; R).
The reason that this is of interest is the following : let X be a G-space and let F be the G-fixed point set of X.
There is a natural map
∗
∗
HG
(X) −→ HG
(F ) = H ∗ (F ) ⊗ H ∗ (BG )
where the target is big and this a map of H ∗ (BG )-modules.
Equivariant deRham Theory
Of course, there is the question of why?
(1) Usual advantages of differential forms - hands-on geometry; local and explicit.
(2) Localization formulas.
(3) Chern-Weil Theory - equivariant characteristic classes in terms of curvature.
(4) Morse theory.
Let’s just do some basic differential topology. Let X be a C ∞ -manifold. There is a natural group Diff(X) of C ∞ diffeomorphisms of X. There is an associated Lie algebra V (X) of C ∞ -vector fields on X. In some sense, V (X)
is the Lie algebra of Diff(X). Given V ∈ V (X) there exists (for some time if X isn’t compact) {ϕt }t∈R ∈ Diff(X)
such that
d
V (x) = ϕt (x) .
dt
t=0
(1) Diff(X) acts on the deRham complex : if ϕ ∈ Diff(X) and α ∈ E k (X) then ϕ · α := (ϕ−1 )∗ α. Furthermore,
ϕ · (ψ · α)
=
ϕ(dα)
=
ϕ(α ∧ β)
=
(ϕ · ψ) · α
d(ϕ · α)
ϕ(α) ∧ ϕ(β).
(2) Taking derivatives with ϕt
L
V (X) −→ Der(E ∗ (X))
is defined by
LV (α) :=
It satisfies
d
d
ϕt α
= ϕ∗t (α) .
dt
dt
t=0
t=0
LV (α ∧ β)
LV (dα)
LV LW − L W LV
= LV (α) ∧ β + α ∧ LV (β)
= d(LV α)
= L[V,W ] .
(3) The contraction gives us a map
ι
It has the following properties :
V (X) −→ Skew Der−1 (E ∗ (X)).
ιV (α ∧ β)
ιV ιW + ιW ιV
LV ιW − ιW LV
d ◦ ιV + ιV
◦
= ιV (α) ∧ β + (−1)|α| α ∧ ιV (β)
=
0
= ι[V,W ]
d = LV .
33
Let G be a compact Lie group with g its Lie algebra. If G acts on X then there is a continuous group
homomorphism G → Diff(X) with an associated Lie algebra homomorphism g → V (X), i.e., given v ∈ g we set
d
ṽ(x) := exp(tv) · x .
dt
t=0
The whole package of operators and relations pull back to (G, g), i.e., G acts on E ∗ (X), L : g → Der(E ∗ (X)) and
ι : g → Skew Der−1 (E ∗ (X)). Moreover, for any v, w ∈ g we have
Lv ◦ d
[Lv , Lw ]
ιv ιw + ιw ιv
Lv ιw − ιw Lv
d ◦ ιv + ιv ◦ d
g ◦ Lv ◦ g
g ◦ ιv ◦ g
−1
−1
= d ◦ Lv
= L[v,w]
=
0
= ι[v,w]
= Lv
= LAdg (v)
= ιAdg (v) .
All of this is nice and can be packaged neatly using graded algebras.
Definition 73. A Z2 -graded algebra (or a superalgebra) is an algebra A with a direct sum decomposition A = A0 ⊕A1
such that
Ai · Aj ⊆ Ai+j mod 2 .
A Z-graded algebra is an algebra A with a direct sum decomposition A = ⊕k∈Z Ak such that
Aj · Al ⊆ Aj+k ∀ j, k ∈ Z.
Notice that any Z-graded algebra is a Z2 -graded algebra by letting A = Aeven ⊕ Aodd .
Example 74. Let V = V0 ⊕ V1 be a super vector space. Then A = End(V ) is a superalgebra with a splitting of
morphisms that preserve the components of V or switches them completely.
Definition 75. Let A be a Z2 (or Z)-graded associative algebra. Then the supercommutator of a, b ∈ A is
[a, b] := ab − (−1)deg a deg b ba
if a, b have pure degree. This extends uniquely to A × A → A. Moreover, A is supercommutative if [·, ·] ≡ 0.
In the Z-graded case this is an example of the following :
Definition 76. A Z-graded super Lie algebra is a Z-graded vector space
g = ⊕∞
k=−∞ gk
with a bilinear bracket
[·, ·] : g × g −→ g
such that
[v, w]
=
(−1)deg v deg w [w, v]
[u, [v, w]]
=
[[u, v], w] + (−1)deg u deg v [v, [u, w]].
Definition 77. Let A be a Z or Z2 -graded algebra. Let End(A) = ⊕Endk (A) where
Endk (A) = {L | L(Aj ) ⊆ Aj+k }.
Set Derk (A) ⊆ Endk (A) consisting of L ∈ Endk (A) such that
L(ab) = L(a)b + (−1)kj aL(b) ∀ a ∈ Aj , b ∈ A.
34
Example 78. If A = E ∗ (X) then d : E ∗ (X) → E ∗ (X) can be thought of as d ∈ Der1 (E ∗ (X)).
Example 79. Let g be a super Lie algebra and fix V ∈ gk . Define LV (W ) := [V, W ]. Then LV ∈ Derk (g) and the
fact that it’s a derivation is equivalent to the super Jacobi identity.
Example 80. Let us go back to the previous setup of G acting on X with g → V (X). The operators d, LV and ιV
are all derivations of E ∗ (X) of degrees 1, 0 and −1 respectively. Define g? := g−1 ⊕ g0 ⊕ g1 where
g−1
g0
g1
≡
≡
≡
{ιV | V ∈ g}
{LV | V ∈ g}
{td | t ∈ R}.
The long list of identities makes g? a super Lie algebra. In other words, whenever X is a G-manifold, G acts on
E ∗ (X) and there is a map of super Lie algebras
g? −→ Der? (E ∗ (X)).
35
Lecture 12 - 10th March ’11
Let A be a graded algebra.
Lemma 81. Der A is a Lie superalgebra.
Proof
Let L1 , L2 ∈ Der A with deg L1 = n1 , deg L2 = n2 and deg a = m. Then
L1 L2 (ab)
(L1 L2 a)b + (−1)mn2 L1 (a)L2 (b)
=
+ (−1)n1 (m+n2 ) L2 (a)L1 (b) + (−1)m(n1 +n2 ) aL1 L2 b
L2 L1 (ab)
(L2 L1 a)b + (−1)n2 (m+n1 ) L1 (a)L2 (b)
=
+ (−1)n1 m L2 (a)L1 (b) + (−1)m(n1 +n2 ) aL2 L1 b
L1 L2 (ab) − (−1)n1 n2 L2 L1 (ab)
=
[L1 , L2 ] (ab)
=
(L1 L2 − (−1)n1 n2 L2 L1 )(a) · b
+ (−1)m(n1 +n2 ) a(L1 L2 − (−1)n1 n2 L2 L1 )b
([L1 , L2 ]a)b + (−1)m(n1 +n2 ) a([L1 , L2 ]b).
This completes the proof.
Observe that any L ∈ Der A is determined by its values on a set of multiplicative generators. if two elements
L1 , L2 agree on a set of generators then L1 ≡ L2 . There is also a notion of tensor products of graded algebras, viz.,
for graded algebras A and B, A ⊗ B is graded algebra where
0
(a ⊗ b)(a0 ⊗ b0 ) := (−1)(deg b)(deg a ) (aa0 ) ⊗ (bb0 )
for elements of pure degree.
Fix a compact Lie group G with Lie algebra g. If G acts on a manifold X then G acts on (E ∗ (X), d) and we
get maps g → Der(E ∗ (X)) induced by L and ι. Define a Lie superalgebra g? := g−1 ⊕ g0 ⊕ g1 where
g−1
g0
g1
≡
{ιV | V ∈ g}
≡
{td | t ∈ R}.
≡
{LV | V ∈ g}
with Lie superbracket
[ιv , ιw ]
[Lv , Lw ]
[d, d]
[Lv , ιw ]
[Lv , d]
[ιv , d]
=
0
= L[v,w]
=
2d2 = 0
= ι[v,w]
=
0
= Lv .
Notice that now all the signs are gone!
Definition 82. Let A be a graded algebra. It is called a G-algebra if there is a representation
ϕ : G −→ Aut A
and a representation (i.e., homomorphism of Lie superalgebras)
L : g? −→ Der A.
The same thing works for modules too.
Example 83. The fundamental example is that of G acting on a manifold X. What about morphism?
36
Definition 84. Let A, A0 be G-algebras. A morphism of G-algebras is an ordinary homomorphism of algebras
F : A → A0 which is
(i) G-equivariant
(ii) Lv ◦ F = F ◦ Lv
(iii) ιv ◦ F = F ◦ ιv
(iv) d ◦ F = F ◦ d.
If f : X 0 → X is a G-map between G-manifolds then
F := f ∗ : E ∗ (X) −→ E ∗ (X 0 )
is a morphism of G-algebras following the definition above.
Let A be a G-algebra. Then H(A) := ker d/Im d is the (co)homology of A.
Definition 85. We call A acyclic if H(A) = 0.
Let’s look at the fundamental example again - here A = E ∗ (X) and
H(A) = HdR (X) ∼
= H ∗ (X; R).
Suppose G acts freely on X. What is the de Rham analogue of H ∗ (X/G)?
Let π : X → X/G be the principal bundle. Then π ∗ : E ∗ (X/G) → E ∗ (X) is a map of complexes. It is
inherently injective and I’ll leave it as an exercise to show that π ∗ (E ∗ (X/G)) consists of all forms α ∈ E ∗ (X) such
that it satisfies
(i) g ∗ α = α for any g ∈ G,
(ii) ιv α = 0 for any v ∈ g.
This motivates the following :
Definition 86. Given a manifold X with G acting on it, we define
∗
Ebasic
(X) := {α ∈ E ∗ (X) | (i), (ii) holds}
We define the basic cohomology to be
∗
∗
Hbasic
(X) := H ∗ (Ebasic
(X)).
Now we pass to algebra.
Definition 87. Let V be a G-module. Define
Vbasic := {x ∈ V | gx = x ∀ g ∈ G, ιv x = 0 ∀ v ∈ g}.
Since d commutes with g and ιv , Vbasic is a subcomplex of V under d. We may therefore define Hbasic (V ) :=
H(Vbasic , d).
Definition 88. Let A, B be G-algebras (or G-modules). A chain homotopy from A to B is a G-equivariant map
T : A → B of odd degree (of degree −1 for Z-graded algebras) such that for any v ∈ g
(i) ιv ◦ T + T ◦ ιv = 0
(ii) Lv ◦ T − T ◦ Lv = 0.
Given such a T we set τ := [T, d] = d ◦ T + T ◦ d.
Lemma 89. τ is a morphism of G-algebras (or G-modules).
Proof
Since T and d are G-equivariant, so is τ . It also commutes with d. One can check that
ιv ◦ τ
= ιv ◦ d ◦ T + ιv ◦ T ◦ d
= L v ◦ T − d ◦ ιv ◦ T + ιv ◦ T ◦ d
= T ◦ Lv − d ◦ ιv ◦ T − T ◦ ιv ◦ d
= T ◦ d ◦ ιv − d ◦ ιv ◦ T
= τ ◦ ιv .
As Lv commutes with T (by (ii)) and with d, we have Lv ◦ τ = τ ◦ Lv .
37
Proposition 90. Let f0 , f1 : A → B be morphism of G-algebras (or G-modules). Suppose they are chain homotopic, i.e.,
there is a chain homotopy T : A → B such that f1 − f0 = τ = d ◦ T + T ◦ d. Then
(f0 )∗ ≡ (f1 )∗ : H(A) −→ H(B).
Proof
Take a ∈ A with da = 0. Since
f1 (a) − f0 (a) = d ◦ T (a) + T (da) = d ◦ T (a)
this implies (f1 )∗ [a] = (f0 )∗ [a]. It can be checked that if a ∈ Abasic then T a ∈ Bbasic .
0
0
We get back to the fundamental example again - Let f0 , f1 : X → X be G-maps. Let F : X × [0, 1] → X be a
G-homotopy. This gives a chain homotopy from f1∗ to f0∗ . More precisely, if F ∗ : E ∗ (X) → E ∗ (X 0 × [0, 1]) and
pr
X 0 × [0, 1] −→ X 0 then T := pr∗ ◦ F ∗ is the prescribed chain homotopy. One can think of T as integrating over
the fibre.
38
Lecture 13 - 15th March ’11
We need to discuss the notion of G-algebras with connections. In the fundamental case G acts on a manifold
X inducing a map
g −→ V(X).
Suppose G acts freely. Then this gives a subbundle V ⊂ T X given by
Vx := {ṽ(x) | v ∈ g} = image of g.
V is a G-invariant subbundle of T X. Now choose7 a G-invariant subbundle V ∗ ⊂ T ∗ X such that
V ∗ × V −→ R
is non-degenerate, whence g∗ ∼
= V ∗.
For example, suppose X has a G-invariant metric. Then
= V ⊕V⊥
TX
T ∗X
= V ∗ ⊕ (V ⊥ )∗ = V ∗ ⊕ (V ∗ )⊥ .
This gives a G-equivariant map
ω
g∗ −→ E 1 (X).
(2)
For ξ ∈ g∗ and v ∈ g one has
ιṽ ω(ξ) = ξ(v).
Customarily in differential geometry one discusses the adjoint ω : T X → g.
Choose a basis v1 , . . . , vN of g. Get a dual basis ξ1 , . . . , ξN of g∗ . We have the maps
vj
ξj
7→ ṽj ∈ Γ(V )
7→ ω(ξj ) =: ω j ∈ Γ(V ∗ ) ⊂ E 1 (X).
We shall adopt the following notation :
Lṽk ≡ Lk ,
ιṽk ≡ ιk , [vk , vj ] =
N
X
cm
kj vm .
m=1
Since ιk ω l = δkl we see that
0
= Lj (ιk ω l )
= ιk Lj ω l + ι[vj ,vk ] ω l
X
l
= ιk L j ω l +
cm
jk ιm ω
= ιk Lj ω l + cljk .
Since V ∗ is G-invariant
Lj ω l
=
l
=
ιk Lj ω
N
X
aljk ω k
k=1
−cljk
= aljk .
7 I emphasize the fact that there is no way out of this choice. We are choosing a connection and sometimes it is possible to choose a connection
with some extra structure.
39
Therefore, we get
Lj ω l
= −
ιj dω l
= −
Write
Ωl ≡ dω l +
N
X
clkj ω k
k=1
N
X
cjk ω k .
k=1
N
1 X l j
cjk ω ∧ ω k .
2
j,k=1
We clain that Ωl is horizontal, i.e., ιj Ωl ≡ 0 for any j. Notice that
ι j Ωl
= ιj {dω l +
= −
= −
=
0.
N
X
N
1 X l j
cjk ω ∧ ω k }
2
j,k=1
cljk ω k +
k=1
N
X
k=1
1X l
ckm ιj ω k ∧ ω m − ω k ∧ ιj ω m
2
k,m
1
1
cljk ω k + cljm ω m − clkj ω k
2
2
If one studies (2) closely then it is clear that
(3)
1
Ω = dω + [ω, ω]
2
This is called the Maurer-Cartan equation.
Now let us reformulate this in an algebraic context.
Definition 91. A G-algebra with connection is a pair (A, ω) where A is a G-algebra and
ω : g∗ −→ A1
is a G-equivariant linear map such that
ιv ω(ξ) = ξ(v) ∀ v ∈ g, ∀ ξ ∈ g∗ .
Choose a basis v1 , . . . , vN of g. Get a dual basis ξ1 , . . . , ξN of g∗ . Set ω l ≡ ω(ξl ) ∈ A1 for l = 1, . . . , N . Therefore,
for ω 1 , . . . , ω N ∈ A1 satisfy
ιj ω l
=
δkl
Lj ω l
=
−
ιj dω l
=
−
N
X
cjk lω k
k=1
N
X
cljk ω k .
k=1
Define Ωl as before and get a map ω : A∗ → g with curvature Ω.
The Weil Algebra
40
We want a minimal G-algebra with connection which is acyclic and freely generated. It will map into any (A, ω)
uniquely.
Let G, g be fixed. Consider the differential graded algebra
W ≡ Λ∗ g∗ ⊗ S ∗ g∗ = Λ∗ ⊗ S ∗
with deg(Λk ⊗ 1) = k and deg(1 ⊗ S k ) = 2k. The differential is given by
d : λ1 ⊗ 1 −→ 1 ⊗ S 1 , d(u ⊗ 1) = 1 ⊗ u.
Since d2 = 0 we have to define d(1 ⊗ S 1 ) ≡ 0. Now extend d as (super)derivation to all of W. G acts on W - G
acts on g by the Adjoint action and it acts on g∗ by the coAdjoint action, i.e., if ω ∈ g∗ then
g · ω = (Adg−1 )∗ ω.
Given v, v 0 ∈ g we get
(Lv ω)(v 0 )
=
=
d
ω(Adexp(−tv) v 0 )
dt
t=0
−ω([v, v 0 ]).
Choose a basis (and a dual basis) of g as before.
(Lvi ω k )(vj )
−ω k ([vi , vj ])
X
−ω k
clij vl
=
=
l
−ckij .
=
On W we have g∗ = Λ1 ⊗ 1 with basis ω 1 , . . . , ω N and g∗ = 1 ⊗ S 1 with (same dual basis) basis x1 , . . . , xN . These
have to satisfy the identities :
dω k
= xk
dxk
=
Li ω
k
0
= −
Li xk
= −
ιi ω k
= δik
k
ιi x
= −
N
X
ckij ω j
j=1
N
X
ckij xk
j=1
N
X
ckij ω j .
j=1
These equations give all the relations of a G-algebra (which are equalities of derivations) on the generators and since
W is generated as an algebra by Λ1 ⊗ 1 and 1 ⊗ S 1 , we’re done.
Theorem 92. W is a G-algebra with connection.
The next thing to observe is that the Weyl algebra is acyclic.
Lemma 93. W is acyclic.
Proof
Define T : W → W of degree −1 given by
T (1 ⊗ xk ) = ω k ⊗ 1, T (ω k ⊗ 1) = 0
41
and extended as a derivation. Now
(d ◦ T + T ◦ d)(ω k ⊗ 1) = T (dω k ⊗ 1) = T (1 ⊗ xk ) = ω k ⊗ 1.
Similarly, [d, T ] is the identity on 1 ⊗ S 1 . Therefore,
[d, T ] = (k + l)Id on Λk ⊗ S l .
T is a chain homotopy to “Id" and hence H(W, d) = 0.
Note T is not a G-algebra morphism. In fact, ιv ◦ T + T ◦ ιv 6= 0. Moreover, HG (W = Hbasic (W) 6= 0. We will
see that
HG (W) = HG (pt) ∼
= S ∗ (g∗ )G .
42
Lecture 14 - 22nd March ’11
I had started to talk about the Weil algebra. Let me remind you again what it is.
Definition 94. (Weil algebra)
Let G be a compact Lie group with Lie algebra g. Define the Weil algebra
W := Λ∗ g∗ ⊗ S ∗ g∗
where Λ1 g is g with a basis ω 1 , . . . , ω N and S 1 g = g∗ with basis x1 , . . . , xN (same basis). The various operators
are defined by
d(xk )
=
0
d(ω )
=
xk
Lvi (ω k )
=
−
Lvi (xk )
=
k
k
−
N
X
ckij ω j
j=1
N
X
ckij xj
j=1
ιvi (ω )
=
δik
ιvi (xk )
=
−
N
X
ckij ω j .
j=1
The identities above imply that W is a G-algebra. Moreover, these hold on generators and by Leibnitz they hold
on all of W. It is a G-algebra with a connection. We had seen last time that W is acyclic.
We define
N
1X k i
k
k
c ω ∧ ω j , k = 1, . . . , N.
Ω := x +
2 j=1 ij
Proposition 95. For the Weil algebra W = Λ∗ g∗ ⊗ Whor where
Whor = C[Ω1 , . . . , ΩN ].
Proof
We see that
N
ιvl (Ωk )
= ιvl (xk ) +
= −
= −
=
0.
N
X
1X k
c ιv (ω i ∧ ω j )
2 j=1 ij l
cklj ω j +
j=1
N
X
N
1 X k
c (δil ω j − δjl ω i )
2 i,j=1 ij
N
cklj ω j +
j=1
N
1X k j 1X k i
c ω −
c ω
2 j=1 lj
2 i=1 il
This implies that Ωk is horizontal, whence
C[Ω1 , . . . , ΩN ] ⊆ Whor .
Since (Λ∗ g∗ ) ∩ Wbasic = {0} (this is because for any ϕ ∈ Λ∗ g∗ ⊗ 1 one can find v ∈ g such that ιv ϕ 6= 0).
W = Λ∗ g∗ ⊗ C[Ω1 , . . . , ΩN ] and conclusion?
43
Corollary 96. The basic algebra is given by
Wbasic = C[Ω1 , . . . , ΩN ]G ,
also thought of as the space of G-invariant polynomials in Ω1 , . . . , ΩN .
Observe that the basic algebra Wbasic is d-invariant and d has degree 1, whence d ≡ 0 on Wbasic (since all elements
are of even degree). This implies that
Hbasic (W) = C[Ω1 , . . . , ΩN ]G = S ∗ (g∗ )G .
We will now prove that this Weil algebra is universal.
Theorem 97. For every G-algebra with connection (A, ω) there exists a G-morphism
ϕ : W −→ A
and a homomorphism
ϕ∗
S ∗ (g∗ )G = Hbasic (W) −→ Hbasic (A).
Furthermore, if ω
e is another connection on A with an associated map ϕ
e : W → A then
ϕ
e∗ = ϕ∗ : Hbasic (W) −→ Hbasic (A).
Definition 98. The map ϕ is called the Chern-Weil map and ϕ∗ is called the Chern-Weil homomorphism.
Let’s look at the fundamental example. Let X be a smooth principal G-bundle. Set A = E ∗ (X). Given a connection
ω : g∗ → E 1 (X) we can think of it dually as a G-equivariant map ω̃ : T X → g. By the theorem, there is a
homomorphism of G-algebras
ϕ
W −→ E ∗ (X).
In particular,
π∗
ϕ
even
Wbasic = C[Ω1 , . . . , ΩN ]G −→ Ebasic
(X) −→ E even (X/G).
Therefore, Im ϕ ⊂ E even (X/G) consists of closed forms canonically chosen by the connection ω.
/ E even (X/G)
S ∗ (g∗ )G
MMM
MMMϕ∗
MMM
MM& H even (X/G)
ϕ
Chern-Weil theory says that ϕ∗ is independent of the connection chosen.
Proof
Given (A, ωA ) we define a map
g∗ −→ A1 , ω 7→ ωA
given by ω(v) = ιv (ΩA ). This determines a linear map on the free generators of W which extends uniquely to a
G-algebra homomorphism ϕ : W → A.
Now suppose that ωA , ω
eA are two connections on A. Set
ωt,A := (1 − t)ωA + te
ωA , t ∈ [0, 1].
This is a connection for each t. So we get
Define Ṫt : W → A of degree −1 by
ϕt : W −→ A.
Ṫt (ω k )
=
Ṫt (dω k )
=
44
0
d
ϕt (ω k )
dt
and extend it8 as a derivation. It is clear that Ṫt is G-equivariant. We claim that
Ṫt ιv + ιv Ṫt = 0 ∀ v ∈ g.
This follows from
(Ṫt ιv + ιv Ṫt )(ω k )
k
(Ṫt ιv + ιv Ṫt )(dω )
=
Ṫt (ιv (ω k ))
=
Ṫt (δ·v )
=
0
= Ṫt (ιv (dω k ) + ιv (Ṫt (dω k ))
= Ṫt (−dιv (ω k ) + Lv ω k ) + ιv (
=
Lv (Ṫt ω k ) +
=
0.
d
ϕt (ω k ))
dt
d
ϕt (ιv ω k )
dt
Therefore, Ṫt is a chain homotopy for all t. Now
d
ϕt (ω k )
dt
d
(dṪt + Ṫt )(dω k ) = d
ϕt (ω k )
dt
d
=
ϕt (dω k ).
dt
(dṪt + Ṫt )(ω k )
Define
T :=
=
Z
1
Ṫt dt.
0
It is G-equivariant map satisfying T ιv + ιv T = 0 satisfying
d ◦ T + T ◦ d = ϕ1 − ϕ0 = ϕ
e − ϕ.
This proves the independence of the choice of the connection.
8 Notice
that we actually need to have an algebra with a topology to take limits but in the cases we’ll be interested in this works fine.
45
Lecture 15 - 24th March ’11
I had started to talk about the Weil algebra. I wanted to make an observation. Recall that
Whor = C[Ω1 , . . . , ΩN ] ⊂ W
and d 6= 0 on Whor . It’s worth computing what d is.
dΩl
=
=
=
=
=
1X l i
cij ω ∧ ω j
d dω l +
2 i,j
1X l
cij dω i ∧ ω j − ω i ∧ dω j
2 i,j
o
1 l n i 1 X i
1X j
cij Ω −
ckm ω k ∧ ω m ∧ ω j − ω i ∧ Ωj −
ckm ω k ∧ ω m
2
2
2
k,m
k,m
o
n
1X l i
(The other terms are zero by Jacobi)
cij Ω ∧ ω j − clij ω i ∧ Ωl
2 i,j
X
−
clij ω i ∧ Ωj .
i,j
The identity above is called the Bianchi identity. Recall that
X
Li Ωl = −
clij Ωj .
j
Therefore,
dΩl =
N
X
k=1
ω k ∧ L k Ωl .
Notice that if p(Ω , . . . , Ω ) ∈ C[Ω , . . . , Ω ] then Lk (p(Ω)) = 0 for any k due to G-invariance, whence p(Ω)
is closed.
Let W be the universal G-algebra with a connection
1
N
1
N G
g∗ −→ g∗ ⊗ 1 ⊂ W.
Fix a basis v1 , . . . , vN of g with a dual basis ω 1 , . . . , ω N of g∗ . Define
X
ω =
ω k ⊗ vk ∈ g∗ ⊗ g
k
Ω
=
X
k
Ωk ⊗ vk ∈ W ⊗ g.
Then we have the Maurer-Cartan and Bianchi identities :
Ω
dΩ
1
= dω + ω, ω
2
= − ω, Ω .
We’re heading towards Mathai-Quillen theorem. The idea at this point is to define
HG,deR (X) := Hbasic (E ∗ (X) ⊗ E)
given a G-manifold X and any acyclic G-algebra E with a connection. We then have to show independence of E
and equivalence with HG (X). Then we have these models :
46
The Weil Model
The Cartan Model
HG (X) ∼
= Hbasic (E ∗ (X) ⊗ W).
HG (X) = H (E ∗ (X) ⊗ S ∗ (g∗ ))G , dCar .
This is a much simpler model but the differential is twisted.
The Stiefel Model
If G ⊂ Un then we define
HG (X) = lim Hbasic (E ∗ (X × Stn,N )).
N →∞
Definition 99. A W-module is a G-module which is also a W-module such that the module multiplication map
W ⊗ A −→ A
is a G-morphism.
One can similarly define a W-algebra. Perhaps it’s worth writing down the identities involved :
Lk φ(ω ⊗ a)
ιk (ω ⊗ a)
dφ(ω ⊗ a)
= φ(Lk ω ⊗ a + ω ⊗ Lk a)
= φ(ιk ω ⊗ a + (−1)deg ω ω ⊗ ιk a)
= φ(dω ⊗ a + (−1)deg ω ω ⊗ da).
Notice that a W-algebra is exactly a G-algebra with a connection.
Fix W with distinguished generators
ω 1 , . . . , ω N , Ω1 , . . . , ΩN
corresponding to a choice of basis v1 , . . . , vN of g. Let A be a W-algebra and let B be a G-module.
Definition 100. We define τ ∈ End(A ⊗ B) by
τ :=
N
X
k=1
It has the following properties :
(1) τ is independent of choice of basis.
(2) τ is G-equivariant.
(3) τ N +1 = 0.
ω k ⊗ ι vk .
It should be clear that (1) holds. For (2), let g ∈ G.
X
g −1 τ g(a ⊗ b) =
(g −1 ω k ga) ⊗ (g −1 ιk gb)
k
=
X
l
(Adtg )−1
kl (Adg )km (ω a) ⊗ (ιm b)
k,l,m
=
X
k,l,m
=
δlm (ω l a) ⊗ (ιm b)
X
(ω l a) ⊗ (ιl b)
l
=
Define
τ (a ⊗ b).
τ3
τ2
+
+ ···
2!
3!
= exp(−τ ), whence Φ ∈ GL(A ⊗ B).
Φ := exp(τ ) = 1 + τ +
It has an inverse given by Φ−1
47
Theorem 101. (Mathai-Quillen)
The operator Φ obeys the following identities :
Φ ◦ (1 ⊗ ιv + ιv ⊗ 1) ◦ Φ−1
(4)
= ιv ⊗ 1
X
X
= d−
Ωk ⊗ ι k +
ω k ⊗ Lk .
Φ ◦ d ◦ Φ−1
(5)
k
k
Moreover, if A and B are W-algebras then Φ is an algebra homomorphism.
Proof This is really a boring calculation. I’ll refer you to Guillemin-Sternberg - Supersymmetry & Equivariant
Cohomology Theory.
By (6)
Φ (A ⊗ B)hor = Ahor ⊗ B.
Since Φ is G-equivariant,
Example 102. Set A = W. Then
Φ (A ⊗ B)basic = (Ahor ⊗ B)G .
G
G
Φ (W ⊗ B)basic = (Whor ⊗ B)G = C[Ω1 , . . . , ΩN ] ⊗ B ∼
= S ∗ (g ∗ ⊗B .
Notice that
dW
By (6) Φ conjugates d to
X
X
d−
Ωk ⊗ ι k +
ω k ⊗ Lk
k
hor
k
dW⊗B
=
k
X
So on (W ⊗ B)G the following identity holds :
X
|
ω k Lk
= dW ⊗ 1 + 1 ⊗ dB .
k
=
X
=
k
ω k Lk ⊗ 1 + 1 ⊗ dB −
X
k
Ωk ⊗ ι k +
X
k
ω k ⊗ Lk
X
Ωk ⊗ ι k .
(ω ⊗ 1) Lk ⊗ 1 + 1 ⊗ Lk +1 ⊗ dB −
k
{z
}
G
vanishes on (W ⊗ B)
Φ ◦ d ◦ Φ−1 = 1 ⊗ dB −
X
k
k
Ωk ⊗ ι k .
Hence, Φ transforms the Weil complex (W ⊗ B)basic , d to the Cartan complex (S ∗ (g∗ ) ⊗ B)G , dCar where
dCar = Φ ◦ d ◦ Φ−1 .
Theorem 103. The Mathai-Quillen
isomorphism Φ carries the Weil complex (W ⊗ B)basic , d to the Cartan complex
(S ∗ (g∗ ) ⊗ B)G , dCar and induces an isomorphism
∗
Hbasic
(W ⊗ B) ∼
= H ∗ (S ∗ (g∗ ) ⊗ B)G , dCar
Note that S ∗ (g∗ ) ⊗ B is the space of all polynomial maps from g to B.
48
Lecture 16 - 29th March ’11
Let’s recall the Mathai-Quillen theorem :
Theorem 104. (Mathai-Quillen)
Let A be a W-module A and B be a G-module with Φ ∈ GL(A ⊗ B) the MQ isomorphism. The operator Φ obeys the
following identities :
(6)
(7)
Φ ◦ (1 ⊗ ιv + ιv ⊗ 1) ◦ Φ−1
Φ◦d◦Φ
−1
= ιv ⊗ 1
X
X
= d−
Ωk ⊗ ι k +
ω k ⊗ Lk .
k
k
Moreover, if A and B are W-algebras then Φ is an algebra homomorphism.
As a consequence,
Φ (A ⊗ B)basic = (Ahor ⊗ B)G .
Consider the special case A = W.
Theorem 105. The Mathai-Quillen
isomorphism
Φ
carries
the
Weil
complex
(W
⊗
B)
,
d
to the Cartan complex
basic
∗ ∗
G
(S (g ) ⊗ B) , dCar and induces an isomorphism
∗
Hbasic
(W ⊗ B) ∼
= H ∗ (S ∗ (g∗ ) ⊗ B)G , dCar
Note that S ∗ (g∗ ) ⊗ B is the space of all polynomial maps from g to B. Let P : g → B be such a map. Then
for any v ∈ g,
(dCar P)(v) = dB (P(v)) − ιv P(v).
To see this, fix a basis v1 , . . . , vN of g and a dual basis x1 , . . . , xN of g∗ . Think of xk as a real valued linear function
on g. We have
S ∗ g∗ ∼
= C[x1 , . . . , xN ]
and elements of S ∗ g∗ ⊗ B are finite sums of elements of the form
X
aα xα ⊗ b, b ∈ B.
p(x) ⊗ b =
Here we are using the multi-index notation : xα = (x1 )α1 · · · (xN )αN . Since
dCar = 1 ⊗ dB −
N
X
xk ⊗ ι k ,
=
X
k=1
we compute
X
k
k
x ⊗ ιk
X
α
aα x
α
⊗b
k
=
XX
α
k
=
X
α
=
X
α
=
49
xk
X
α
aα xα ⊗ ιk b
aα xα ⊗ (xk ιk b)
aα xα ⊗ ιPk xk vk b
aα xα ⊗ ιX b
(1 ⊗ ιX )(p(X) ⊗ b).
There
way to look at this. Think of p ∈ (S ∗ g∗ ⊗ B)G as polynomials on g with coefficients in B, i.e.,
Pis another
α
p = bα x with bα ∈ B. Then
X
X
dCar p =
1 ⊗ dB −
xk ⊗ ι k
bα xα
=
X
α
α
k
X
dB (bα )x −
(ιk bα )xk xα
α
Xn
= dB ◦ p(x) −
α
k
o
ιPk xk vk bα xα
= dB ◦ p(x) − ιX p(x).
Recall that in the fundamental case, X is a G-manifold, B = E ∗ (X) and S ∗ g∗ ⊗E ∗ (X)
polynomial maps g → E ∗ (X) with dCar = d − ιṽ .
G
is the space of G-invariant
Theorem 106. (The independence theorem)
Let A be a W-algebra and E be an acyclic W-module. Then
Proof
H ∗ (A ⊗ E)basic ∼
= H ∗ (Abasic ).
Apply the Mathai-Quillen isomorphism
∼
=
Φ : (A ⊗ E)basic −→ (Ahor ⊗ E)G .
So the complex ((A ⊗ E)basic , dA ⊗ 1 + 1 ⊗ dE ) becomes ((Ahor ⊗ E)G , d). We write
X
X
ω k ⊗ Lk −
Ωk ⊗ ι k .
d = 1 ⊗ dE + dA ⊗ 1 +
| {z } |
{z
}
d
1
d2
Set C k := (Ahor ⊗ E k )G . Then
(Ahor ⊗ E)G ≡ C = C 0 ⊕ C 1 ⊕ C 2 ⊕ · · ·
with d1 : C k → C k+1 for any k.
Proposition 107. We have
H k (C, d1 ) =
Proof
Abasic
0
if k = 0
if k > 0.
Consider the full complex
with the same grading. Clearly,
e = Ahor ⊗ E ⊃ (Ahor ⊗ E)G =: C
C
e d1 ) =
H (C,
k
In general, we get
Ahor
0
if k = 0
if k > 0.
e d1 ) = Ahor ⊗ H ∗ (E, dE )
H ∗ (C,
e k with d1 α = 0.
where H ∗ (E, dE ) is concentrated in degree 0 due to the acyclicity of E. Now suppose α ∈ C k ⊂ C
k
If k = 0 then α ∈ Ahor and α is G-invariant and therefore α ∈ Abasic . If α ∈ C (k > 1) then α = d1 β̃ for some
e k−1 . Set
β̃ ∈ C
Z
β :=
G
(g · β̃)dg
50
e k−1 )G =: C k−1 .
using the Haar measure. Then α = dβ with β ∈ (C
We now filter our complex C as follows :
C 0 ⊂ (C 0 ⊕ C 1 ) ⊂ (C 0 ⊕ C 1 ⊕ C 2 ) ⊂ · · ·
|{z}
| {z } |
{z
}
F0
F1
F2
with d1 : C k → C k+1 and d2 : C k → C k ⊕ C k+1 . Notice that this means d1 : F k → F k+1 and d2 : F k → F k .
Proceed by induction on k to show that
H ∗ (F k ) = H ∗ (Abasic )
which completes the proof.
0
Let k = 0 and given a ∈ C with da = 0. Then d1 a = 0 and
a ∈ ker d1 ∩ C 0 = AG
hor = Abasic .
In other words, a defines a class in H ∗ (Abasic , d). Assume that this is true for k −1. Given a ∈ F k such that da = 0,
e k−1 such that a = d1 b. Set
we have d1 a = 0. By Proposition 107 there exists b ∈ C
ã = a − db = a − d1 b − d2 b = −d2 b ∈ F k−1 .
By construction, dã = 0 and we apply induction.
51
Lecture 17 - 31st March ’11
Let me remind you of what I was trying to do last time. We had a filtration
F1 ⊂ F2 ⊂ · · ·
of C ∗ . The inductive statement was that given a ∈ F k such that da = 0 there exists b ∈ F 1 such that
a + db ∈ Abasic ⊂ F 0 .
Step 1 : da = 0 ⇒ d1 a = 0 ⇒ a ∈ AG
hor
Step 2 : missed something
Theorem 108. Let A be a W-module. If E and E 0 are acyclic W-algebras then
Hbasic (A ⊗ E) ∼
= Hbasic (A ⊗ E 0 ).
Corollary 109. For any G-algebra A,
HG (A) := Hbasic (A ⊗ E)
is independent of the choice of E.
Proof
Since
we have
which completes the proof.
(A ⊗ E) ⊗ E 0 ∼
= A ⊗ (E ⊗ E 0 )
Hbasic (A ⊗ E) ∼
= Hbasic (A ⊗ E ⊗ E 0 ) ∼
= Hbasic (A ⊗ E 0 )
Therefore, we can use the model
HG (A) = H (S ∗ g∗ ⊗ A)G , dCar .
If A = E ∗ (X) where X is a G-manifold then this gives a de Rham model for HG (X).
Question
Is this the same as Borel equivariant cohomology?
Theorem 110. Let X be a compact G-manifold. Then there exists a canonical isomorphism
HG,dR (X) ∼
= HG (X).
Proof
Let G ⊂ Un and set Stn,N = UN /UN −n . We have a family
Stn,N
Un
i
/ Stn,N +1
i
/ ···
i
/ ···
Un
Gn,N
i
/ Gn,N +1
of principal Un -bundles (and free G-spaces) such that
πk (Stn,N +1 ) = 0 if k < 2(N − n).
Moreover, the direct limit of Stn,N is contractible. Let EN = E ∗ (Stn,N ). Then
lim H k (EN , d) = 0
N →∞
and E = limN →∞ E ∗ (Stn,N ) is an acyclic G-algebra. There is a natural connection in E which makes it a Walgebra. In fact, each Stn,N carries a canonical UN × G-invariant metric coming from the bi-invariant metric on
UN .
i /
/ ···
Stn,N
Stn,N +1 i
Un
Un
XN
i
/ XN +1
52
i
/ ···
We can get a G-invariant splitting
Tx XN = Vx ⊕ Vx⊥
∼
=
With ix : g → Vx this gives
/g
O
T XN
i−1
=
This is a g-valued 1-form whose adjoint
V ⊕V⊥
pr
/V
g∗ −→ E 1 (XN )
gives a connection. They are compatible under forming E ∗ (X) ⊗ E ∗ (Stn,N ) and taking limits. We denote this by
E ∗ (X) ⊗ E and set
HG,dR (E ∗ (X)) = Hbasic (E ∗ (X) ⊗ E).
Now, we have an inclusion of W-algebras
E ∗ (X) ⊗ E ∗ (Stn,N ) ⊂ E ∗ (X) × Stn,N ).
We claim that this induces an isomorphism in basic cohomology. Therefore,
Hbasic (E ∗ (X) ⊗ E) = lim H∗ ((X × Stn,N )/G) ∼
= H∗ ((X × EG )/G).
N →∞
I wouldn’t prove this claim. Basically it follows from a spectral sequence argument.
From now on we’ll use the Cartan model. Let X be a G-manifold and set
CG (X) := (S ∗ g∗ ⊗ E ∗ (X))G
with dG = d − ιv . More generally, if A is a G-algebra then (CG (A), dG ) is a double complex where CG (A) =
(S ∗ g∗ ⊗ A∗ )G . The differential is given by dG = d1 + d2 where d1 = 1 ⊗ dA and d2 = ιv . We have a bigrading
C p,q := (S p g∗ ⊗ Aq−p )G
with new total degree p + q.
We’re now in the setup of spectral sequence of double complexes. Every double complex (C p,q , d1 , d2 ) has
two spectral sequences associated with it
)
(Er0p,q , δr )
⇒ H total (C total , d1 + d2 ).
00p,q
(Er , δr )
Fact
I’m not going to prove this. It’s very basic and I’ll leave it as an exercise.
E10p,q = (S p g∗ ⊗ H q−p A)G .
Lemma 111. The connected component of the identity G acts trivially on H(A).
Proof
On A
Lk = ιk dA + dA ιk
acts trivially on H(A). if dA a = 0 then Lk a = dA (ιk a).
Proposition 112. If G is connected then
E10p,q = (S ∗ g∗ )G ⊗ H(A).
Theorem 113. If G is connected and H odd (A) = 0 then
HG (A) = (S ∗ g∗ )G ⊗ H(A).
53
Lecture 18 - 5th April ’11
Let me make some comments first :
(1) Homogeneous spaces : For a compact Lie group G and X = G/K we have
HG (G/K) = HK (pt) = (S ∗ K∗ )K .
(2) Let T ⊂ G be a maximal torus.
Theorem 114. (Cartan)
T is unique up to conjugation.
My favourite reference for this is the book by Adams.
Definition 115. Let W = N (T )/T where
N (T ) := {g ∈ G | gT g −1 = T }.
It is called the Weyl group of G.
Theorem 116. Let G/AdG be the orbit space of G acting by conjugation on G. Then
(i) G/AdG ∼
= T /W .
(ii) g/AdG ∼
= t/W .
g
t
W (t)
Theorem 117. We have an isomorphism
S ∗ (g∗ )G ∼
= S ∗ (t∗ )W .
I wouldn’t prove it in this course but in case you haven’t seen it before, it’s good to know this.
Example 118. Let G = Un with Lie algebra
t
C
g = {A ∈ Mn×n
| A = −A}.
Let a maximal torus be given by
with the Lie algebra given by
 iθ
1

 e

T = 


0


 iθ1

t= 


0
···
..
.
···
···
..
.
···
0
eiθn .
0
iθn .
54




 θj ∈ R/Z







 θj ∈ R .



Now W ∼
= Sn is the symmetric group of permutations of (θ1 , . . . , θn ). Then
S ∗ (un )Un
∼
=
∼
=
C[θ1 , . . . , θn ]Sn
C[σ1 , . . . , σn ]
where σk is the kth symmetric polynomial in the variables θi ’s. For example,
= θ1 + . . . + θn ∼
= tr A
∼
= θ1 · · · θn = det A.
σ1
σn
Example 119. Let G = SOn for n = 2m + 1 with Lie algebra
R
g = son = {A ∈ Mn×n
| At = −A}.
Let a maximal torus be given by


 Rθ 1






T = 



 0



Rθ2
..
1
Rθ =
The Lie algebra is given by

0



 λ1













t= 

















.
Rθn
and the 2 × 2 matrices Rθ is given by
− sin θ
cos θ
cos θ
sin θ










 θj ∈ R/Z









0
.

−λ1
0
0
λ2
−λ2
0
..
.
0
λm
−λm
0
0

























= Rm .
λi ∈ R ∼












Let W ⊃ Sn be the permutation of blocks and let ρk be the reflection in the kth R2 in the decomposition
2
R2m+1 = (⊕m
i=1 R ) ⊕ R.
Conjugation by ρk preserves t and W is generated by permutations of the (λ1 , . . . , λm ) and the maps λm 7→ −λm .
Therefore,
C[λ1 , . . . , λm ]W = C[λ1 ,2 , . . . , λ2m ]Sm = C[σ1 , . . . , σm ]
where σi ’s are symmetric polynomials in λ2j ’s.
Example 120. Let G = SOn with n = 2m. ρk has determinant −1 and
t∼
= {(λ1 , . . . , λm ) | λj ∈ R}.
W is generated by Sm and the maps
(λ1 , . . . , λm ) 7→ (λ1 , . . . , −λi , . . . , −λj , . . . , λn ).
As a consequence,
C[λ1 , . . . , λm ]W = C[λ21 , . . . , λ2m , λ1 , . . . , λm ]Sm = C[σ1 , . . . , σm , λ1 , . . . , λm ]/((λ1 · · · λm )2 = σm )
where σk is the kth symmetric polynomial in λ2i ’s.
55
Calculations and examples
Let X be a compact G manifold. Let
G
CG (X) = S ∗ (g∗ ) ⊗ E ∗ (X)
G
with dG := d − ιṽ . Let p ∈ S k (g∗ ) ⊗ E l (X) . There are two ways to think of such an element.
(1) As a G-equivariant homogeneous polynomial map (of degree k) from g to E l (X).
(2) As a degree k polynomial on g with coefficients in E l (X).
0
Degree 0 With this in mind we see that CG
(X) is the space of G-invariant functions on X with dG = d.
0
Moreover, HG (X) = C. This is not terribly interesting.
Degree 1
1
1
CG
(X) is the space of G-invariant 1-form. For ω ∈ CG
(X) the differential is
dG ω = dω − ω(ṽ).
Notice that dω ∈ E 2 (X)G while ω(ṽ) ∈ g∗ ⊗ E 0 (X). Therefore, ker dG consists of closed G-invariant 1-forms,
which are horizontal and
{closed basic 1-forms}
1
.
HG
(X) =
d(E 0 (X)G )
This is slightly interesting if the action of G is not free.
Degree 2
We have
2
CG
(X)
G
G
S 1 (g∗ ) ⊗ E 0 (X) ⊕ S 0 (g∗ ) ⊗ E 2 (X)
G
g∗ ⊗ E 0 (X) ⊕ E 2 (X)G .
=
=
2
(X) with the decomposition as above. for v ∈ g,
Let Φ = −µ + ω ∈ CG
dG Φ
dω − ιṽ ω − d(µ(ṽ)) + ιṽ µ(ṽ)
dω − ιṽ ω + dµ(ṽ)
=
=
E 3 (X)G ⊕ (g∗ ⊗ E 1 (X))G .
∈
Theerefore, Φ is dG -closed is equivalent to
dω
ω(ṽ)
=
0
= −dµ(ṽ) ∀ v ∈ g.
Now µ : g → E 0 (X) is dual to µ̂ : X → g∗ . Given v ∈ g and x ∈ X
µ(v)(x) = µ̂(x)(v).
Let v1 , . . . , vN be a basis of g and let x1 , . . . , xN be coordinates on g, i.e., the dual basis of g∗ . Write v =
and
µ(v)
=
µ̂(v)
=
N
X
k=1
xk fk , fk = µ(vk ) ∈ C ∞ (X)
(f1 (q), . . . , fN (q)), q ∈ X.
In terms of the basis elements, we have
dµ(v)
=
dµ̂(v)
=
N
X
xk dfk
k=1
dfk
(df1 , . . . , dfN )
= −ιvk ω.
56
P
x k vk
If dG Φ = 0 we get a G-map
µ̂ : (f1 , . . . , fN ) : X → g∗
(8)
satisfying dfk = −ιvk ω. This is called the moment map. If ω is non-degenerate then µ is the symplectic moment
map.
Note
If α ∈ g∗ which is G-fixed (for e.g., α = 0). Then (µ̂−1 (α) is G-invariant. We define
X := µ̂−1 (α)/G
to be the (symplectic) reduction of G.
Note Notice that ω is basic on µ̂−1 (α). The G-invariance in clear. Moreover, ṽ is tangent to µ̂−1 (α) for any
v ∈ g and
ιṽ ω = −d(µ(v)) = 0
since µ(v) is constant on this subset. If α is a regular value then X is a submanifold and ω descends to a d-closed
form ω on X (where G acts freely). Moreover, ω is symplectic implies that ω is symplectic too.
Example 121. Let G = U (1) = S 1 act on (X, ω) with ω symplectic and g ∗ ω = ω for any g ∈ S 1 . Let µ̂ : X → R
and take any regular value α. Then
X = µ̂−1 (α)/S 1
is a symplectic manifold with symplectic form ω. It is called the symplectic reduction and is due to some old
colleagues (Marsden and Weinstein) of mine from Berkeley.
57
Lecture 19 - 7th April ’11
Let me talk about moment maps. Consider the special case X = G where G acts by left translations. Given
2
Φ ∈ CG
(G) with dG Φ = 0. Notice that
k
HG
(G) = 0 if k > 0.
1
Therefore, Φ = dG ϕ for ϕ ∈ CG
(G). But
1
CG
(G) = S 0 g∗ ⊗ E 1 (G)
G
= {left invariant 1-forms on g} = g∗ .
Observe that dG ϕ = dϕ − ιṽ ϕ where dϕ is a left invariant 2-form and ιṽ ϕ is a function on G. Let
ṽg =
d
exp(tv) · g t=0
dt
be a right invariant vector field on G which agrees with v at the identity. Think of the linear map
g −→ C ∞ (G), v 7→ ϕ(ṽ)(g), g ∈ G
as a map
G −→ g∗ , g 7→ ϕ(·)(g).
At g ∈ G and v ∈ g
ϕ(ṽ)(g)
= ϕg (ṽg )
= ϕe (Lg−1 )∗ ṽg
Hence, our mapping
= ϕe (Lg−1 )∗ (Rg )∗ ṽe
= ϕe (Adg−1 )∗ v
= Ad∗g−1 ϕe (v).
G −→ g∗ , g 7→ g · ϕ := (Ad∗g−1 )(ϕ)
is the coadjoint action of G on g∗ . This is a G-equivariant map with e 7→ ϕ and
Image = orbit of ϕ ∼
= G/Gϕ ,
where Gϕ := {g | gϕ = ϕ}. Moreover, Gϕ acts on G from the right.
Theorem 122. The left G-invariant 2 form ω := dϕ is basic for right multiplication by Gϕ on G and its image
ω
e ∈ E 2 (G/Gϕ ) is d-closed and non-degenerate. In particular, G/Gϕ is a symplectic manifold.
Proof
˜
Let v ∈ g and let ṽ˜ be the left invariant vector field on G with ṽ(e)
= v. Therefore,
We have the formula
d
ṽ˜g := g · exp(tv)t=0 .
dt
ιṽ˜ ω = ιṽ˜ dϕ = −d(ιṽ˜ ϕ) + Lṽ˜ ϕ = Lṽ˜ ϕ
by left invariance9 . Given v ∈ g, we conclude that
v ∈ gϕ ⇔ Lṽ˜ (ϕ) = 0
where gϕ ⊂ g is the Lie subalgebra of Gϕ ⊂ G. Therefore,
ιṽ˜ ω = 0 ⇔ Lṽ˜ (ϕ) = 0 ⇔ v ∈ gϕ .
(9)
9 It
is true that Lṽ ϕ = 0.
58
Hence, ω = dϕ is horizontal (for right multiplication). Moreover,
Rg∗ ω = Rg∗ dϕ = dRg∗ ϕ = dϕ = ω, g ∈ Gϕ
implies that ω is Gϕ -invariant. In other words, ω is Gϕ -basic. By (9) ω
e on G/Gϕ is non-degenerate.
Let X = G. In degree 2, we get elements G → g∗ and G/Gϕ becomes a symplectic manifold.
Question
In degree 3 what happens? Is it related to equivariant gerbes?
It’s an interesting question that I don’t know the answer to.
Matt’s answer You get group valued moment maps and the usual 3-form you get is the third power of the natural
Maurer-Cartan form on G.
Equivariant Currents
Let X be a compact, oriented G-manifold and dim G = n. Recall that
G
E ∗ (X) = S ∗ (g∗ ) ⊗ E ∗ (X)
with dG = d − ιṽ and deg(S p ⊗ E q ) = 2p + q. We also have
G
0∗
EG
(X) := S ∗ (g∗ ) ⊗ E 0∗ (X)
with the same differential and degree as before. The first model can be thought of as polynomials on g with
coefficients in E ∗ (X) while the latter one is polynomials on g with coefficients in E 0∗ (X). There is a natural
pairing
∗
0∗
EG
(X) ⊗ EG
(X) −→ S ∗ (g∗ ).
Let v1 , . . . , vM ∈ g be a basis with dual basis x1 , . . . , xN such that
S ∗ (g∗ ) ∼
= C[x1 , . . . , xN ].
We write
p(x)
=
X
I
q(x)
=
(q, p)
:=
X
J
X
∗
ωI xI ∈ EG
(X)
0∗
TJ xI ∈ EG
(X)
(TJ , ωI )xI xJ .
I,J
∗
∗
(X) called
At v ∈ g we get (q(v), p(v)). We also have the adjoint of dG : EG
(X) → EG
0∗
∂G : E,∗G (X) −→ EG
(X)
defined by
(∂G q, p) := (q, dG p).
Example 123. Let M k ⊂ X be an oriented G-invariant submanifold of finite volume. Assume that G is orientation
preserving. Consider
G
G
[M ] = 1 ⊗ [M ] ∈ S 0 g∗ ⊗ E 0(n−k) (X) = E 0(n−k) (X) .
Notice that g∗ [M ] = [M ] by G-invariance. Now
∂G [M ]
=
∂[M ] − ιṽ [M ]
∂[M ](β) = [M ](dβ)
ιṽ [M ] (γ) = [M ](ιṽ γ).
59
Note that the last term is zero because ṽ is a vector field tangent to M . Consequently,
∂G [M ] = ∂[M ].
Let me write this extremely important but trivial fact as a lemma :
Lemma 124. Let M k and G be as above. Then
∂G [M ] = ∂[M ].
G
G
Example 125. Let ϕ ∈ E k (X) ⊂ S 0 g∗ ⊗ E 0k (X) . For ψ ∈ E n−k (X),
Z
[ϕ](ψ) =
ϕ ∧ ϕ.
X
∂[ϕ](α)
=
[ϕ](dα)
Z
ϕ ∧ dα
=
X
Z
= (−1)k+1
dϕ ∧ α
X
=
A similar calculation gives us
This shows that
(−1)k+1 [dϕ](α).
ιṽ (α) = (−1)k+1 [ιṽ ϕ](α).
∂G = (−1)k+1 dG on S ∗ (g∗ ) ⊗ E k (X).
We’ll take up operator calculus next time and show that it goes through for equivariant forms and currents.
60
Lecture 20 - 12th April ’11
The oldest reference for equivariant Morse theory is A. G. Wasserman “Equivariant Differential Topology"
Topology vol. 8, pg 127-150 (1969). He was the first one to write about it. One can also check Atiyah-Bott, Bott
and F. Kirwan.
Last time we were looking at
0k
EG
= ⊕2p+q=k S ∗ g∗ ⊗ E 0q (X)
G
.
These should be though of as (G-invariant) homogeneous polynomials of degree p on g with E 0q (X)-coefficients.
There is the pairing
∗
0∗
EG
(X) ⊗ EG
(X) −→ S ∗ (g∗ ).
Example 126. Let M k ⊂ X is a G-invariant oriented submanifold of finite volume. Then
∂G [M ] = ∂[M ].
Hence, if ∂[M ] = 0 then ∂G [M ] = 0.
Example 127. Let ϕ ∈ E n−k (X) with
[ϕ](ψ) :=
Z
X
ϕ ∧ ψ ∈ S ∗ g∗ .
We have
∂[ϕ] = [dϕ] − [ιv ϕ].
0∗
(X × X) we get an operator
Let’s get back to operator calculus. Given T ∈ EG
∗
0∗
T : EG
(X) −→ EG
(X)
given by
T(α) := (pr1 )∗ (pr2 )∗ (α) ∧ T .
G
Specifically, if T ∈ S k g∗ ⊗ E 0n−k (X) then
T : S l g∗ ⊗ E i (X)
The boundary operators are related via
G
G
−→ S k+l g∗ ⊗ E 0i−k (X) .
∂T 7→ ∂T := (−1)q+1 dG ◦ T + T ◦ dG .
A few basic examples are in order :
0
(1) If T = ∆ ⊂ X × X and q = 0 we get T(α) = α, i.e., the canonical inclusion ι : EG (X) ,→ EG
(X).
∗
(2) If T is the graph of ϕ in X × X then T(α) = ϕ (α). It is ∂G -closed.
(3) If (a ⊗ 1) ∈ (S k g∗ ⊗ E 0 (X))G , i.e., a ∈ (S k g∗ )G then T(α) = a · α is the module multiplication.
Equivariant Morse Theory
Let X be a compact G-manifold. Let f : X → R be a G-invariant Morse function. We assume that Cr(f ) is a
subset of the fixed points of the G-action if G is connected. We can construct a G-invariant metric on X and a
G-invariant gradient-like flow which is Morse-Smale. Assume that the metric is flat in a neighbourhood of (v, v),
i.e.,
f (u, v) = c − |u|2 + |v|2 .
0
Get T ∈ E 0n−1 (X) ≡ En+1
(X) given by
T = {(x, ϕt x) | x ∈ X, t ≥ 0} ⊂ X × X.
61
We also have Up , Sp - the finite volume unstable and stable manifolds at p ∈ Cr(f ). Let
X
P (α) := lim ϕ∗t α =
[Up ] × [Sp ].
t→∞
p∈Cr(f )
This is G-invariant since
g(Up × Sp ) = Ugp × Sgp ∀ p ∈ Cr(f ).
Since T is a G-invariant submanifold, ∂T = ∂G T , the original equation ∂T = ∆ − P becomes
∂G T = ∆ − P.
(10)
This implies an operator equation
dG ◦ T + T ◦ dG = I − P.
Therefore, we get
X Z
P(α) =
Up
p∈Cr(f )
0
(X). Notice that
where · is the product in EG
Z
Up
α · [Sp ],
α ∈ (S ∗ g∗ )G .
∗
This follows since P is G-invariant and g[Sp ] = [Sp ] for any g ∈ G. As a consequence, P is a projection of EG
(X)
onto
∗
(S ∗ g∗ )G ⊗ SpanR [Sp ] | p ∈ Cr(f ) = HG
(pt) ⊗ Sf =: SG,f
with dG := 1 ⊗ ∂.
∗
∗
(pt)(X) is a free HG
Theorem 128. Let f be a G-invariant Morse function on a compact G-manifold X. Then HG
module isomorphic to
∗
∗
HG
(X) ∼
(pt) ⊗R H ∗ (X).
= HG
This also works for f : X → R and consider sub-level sets f −1 (−∞, c ] inside X.
Example 129. Consider Cn+1 − {0} → Pn with
f ([z]) =
n
X
|zk |2
k
.
kzk2
k=0
Therefore,
∂f
∂zk
n
=
=
=
X
kz k
zk −
l|zl |2 −
2
kzk
kzk4
l=0
o
X
zk n
2
2
kkzk
−
l|z
|
l
kzk4
n
zk X
2
(k
−
l)|z
|
.
l
kzk4
l=0
We may assume that kzk = 1. Then z is critical if and only if ∂f /∂zk = 0 for any k. Notice that zk 6= 0 and
∂f /∂zk = 0 implies that
n
X
l|zl |2 .
k=
l=0
62
This happens only once. Therefore, the critical points of f are ej := [0 : . . . : 0 : 1 : 0 : . . . : 0] with 1 places at the
jth place. It’s an exercise that f is self-indexing, i.e., Indek (f ) = k.
Notice that f is T -invariant where
T = (S 1 × · · · × S 1 ) /∆ ∼
= (S 1 )n .
{z
}
|
(n+1)−times
The associated flow is given by
(11)
ϕt ([z0 : z1 : · · · : zn ]) = [z0 : et z1 : · · · : ent zn ] = [e−kt z0 : e(1−k)t z1 : · · · : e(k−n)t zn ].
One can check that
Using (11) one concludes that
d
f (ϕt z)t=0 > 0 if z 6= Cr(f ).
dt
lim ϕt [z0 : · · · : zn ] = [0 : · · · : 0 : 1 : 0 : · · · : 0] = ek
t→∞
⇔ zk+1 = · · · = zn = 0, zk 6= 0.
Therefore,
Sek
Uek
Then
P =
n
X
k=0
We conclude that
= P(span{e0 , . . . , ek })
= P(span{ek , . . . , en }).
Pk × Pn−k , Pk · Pn−k = 1.
HT (Pn ) = HT∗ (pt) ⊗ R[u]/ un+1 = R[x1 , . . . , xn ] ⊗ R[u]/ un+1 .
We have a similar picture for Grassmannians. Using the Plücker embedding
Gp (Cn+1 ) ,→ P(ΛpC Cn+1 ), ξ = spanC (v1 , . . . , vp ) 7→ v1 ∧ · · · ∧ vp .
Since T = (S 1 )n+1 acts on Cn+1 , it acts on Gp (Cn+1 ). Then a plane P ∈ Gp (Cn+1 ) is T -fixed if
(z0 , z1 , . . . , eit zk , . . . , zn ) ∈ P ∀ t, k.
Now z ∈ P if and only if ek ∈ P for any zk 6= 0. Then a little work gets us to Schubert cells, flag varieties.
63
Lecture 21 - 14th April ’11
Today I’m going to talk about Morse-Bott functions.
Morse-Bott Functions
Let f : X → R be smooth with X compact.
Definition 130. The function f is a Morse-Bott function if
Cr(f ) = F1 ∪ F2 ∪ · · · ∪ Fv
is a disjoint union of smooth submanifolds (which are assumed connected) with the property that if x ∈ Fj then
the Hessian
Hessx (f )Nx (Fj )
is non-degenerate.
Notice that Tx Fj is in the null space of Hessx (f ) since f is constant on Fj . Therefore, Hessx (f ) descends to
Tx X/Tx Fj ∼
= Nx Fj . At x ∈ Fj , for a suitable basis

We define








Hessx (Fj ) ∼
=







nF
λF
..
.
0
−1
..
.
−1
1
..
.
1
















:= no. of 0’s on the diagonal
=
dim F
=
no. of −1’s
=
λ∗F

0
=
index Hessx (f )
no. of +1’s.
We can write the normal bundle N to F as N = N − ⊕ N + such that the Hessian is negative on N − and positive
on N + .
N = N+ ⊕ N−
N ⊂ Cr(f )
N+
N−
N, N ± are not determined canonically. If we fix a Riemannian metric then they are N (F ) := T (F )⊥ . Moreover,
Hessx (f ) on Nx (F ) has λF negative eigenvalues and λ∗F positive eigenvalues.
64
Theorem 131. Let f : X → R be a Morse-Bott function and F ⊂ X Riem a connected component of Cr(f ). Then there
exists
(a) a metric on N = N (F ),
(b) a splitting N = N − ⊕ N + , and,
(c) a diffeomorphism j : Nε ,→ X of a tubular neighbourhood of the zero-section of N onto a neighbourhood of F in
X such that
j ∗ (f ) = f (F ) − kuk2 + kvk2 for (u, v) ∈ Nx− ⊕ Nx+ , x ∈ F.
Once you have this you can transplant a metric g̃ on N down to a neighbourhood of F . For the given fibre metric
of vertical tangent vectors, use the Riemannian connection to lift gF to horizontal directions. Get a metric on X
such the gradient flow ϕt for ∇f is
ϕt (z, u, v) = (x, e−t u, et v).
For any p ∈ Cr(f )
∪t>0 ϕt Np+ (ε)
≡ ∪t<0 ϕt Np− (ε) .
≡
Up
Sp
With this, we set UF to the union of Up and SF to be union of Sp . We have
/ SF
UF × SF
UF
π0
/ F.
Notice that
UF ×F SF := {((x, y) ∈ UF × SF | πU (x)πS (y) ⊂ X × X
is of dimension nF + λF + λ∗F = n.
Definition 132. A flow ϕt : X → X satisfies the generalized Smale conditions if its fixed point set is a finite union
of submanifolds and for each connected component F of Fix(ϕt ) and any p, q ∈ Fix(ϕt )
Up t SFq , UFp t Sq .
Definition 133. A Morse-Bott function is Morse-Bott-Smale if there exists a good gradient flow ϕt , i.e., adapted in
neighbourhoods of components of Cr(A), which satisfies the generalized Smale conditions.
Recall that we say p < q if there is a piecewise broken flow line from p to q. We have a similar definition for
F < F 0.
Lemma 134. Assume that f is MBS10 with flow ϕt . Then F < F 0 implies λF < λF 0 and λ∗F > λ∗F 0 .
Proof
We have the following picture :
q
F0
SF 0
UF
F
10 We’ll
p
be abbreviating Morse-Bott-Smale by MBS from now on.
65
It follows that
SFq0 t Up
Sq t UFp
⇒
⇒
dim SFq0 + dim Up ≥ n + 1
dim Sq + dim UFp ≥ n + 1.
From this we get λ∗p ≥ λ∗q + 1 and λq ≥ λp + 1.
Caution There exists Morse-Bott function for which no Smale flow exists. Take T ⊂ R with f being the
projection to the z-axis. This satisfies the Smale conditions. Now I perturb the upper piece to make it have 4
critical points. Let’s call it C.
2
C
q
p
It has 2 maximum points and 2 saddle points. This doesn’t satisfy the conditions that we want.
66
3
Lecture 22 - 26th April ’11
Let me remind where we were; it’s been a while. We were talking about Morse-Bott functions. Let f : X →
R be smooth with X compact.
Definition 135. The function f is a Morse-Bott function if
Cr(f ) = F1 ∪ F2 ∪ · · · ∪ Fv
is a disjoint union of smooth submanifolds (which are assumed connected) with the property that if x ∈ Fj then
the Hessian
Hessx (f )Nx (Fj )
is non-degenerate.
Theorem 136. Let f : X → R be a Morse-Bott function and F ⊂ X Riem a connected component of Cr(f ). Then there
exists
(a) a metric on N = N (F ),
(b) a splitting N = N − ⊕ N + , and,
(c) a diffeomorphism j : Nε ,→ X of a tubular neighbourhood of the zero-section of N onto a neighbourhood of F in
X such that
j ∗ (f ) = f (F ) − kuk2 + kvk2 for (u, v) ∈ Nx− ⊕ Nx+ , x ∈ F.
(d) If ϕt (x, u, v) = (x, e−t u, et v) then for p ∈ F let
Up
:= {x | lim ϕ−t (x) = p}
Sp
:= {x | lim ϕt (x) = p}.
t→∞
t→∞
Then we have
SF
=
[
Sp =
p∈F
UF
=
[
[
ϕ−t (Nε− (F ))
t≥0
Up =
p∈F
[
ϕt (Nε+ (F )).
t≥0
Here SF → F and Uf → F are vector bundles.
Definition 137. A flow ϕt : X → X satisfies the generalized Smale conditions if its fixed point set is a finite union
of submanifolds and for each connected component F of Fix(ϕt ) and any p, q ∈ Fix(ϕt )
Up t SFq , UFp t Sq .
Definition 138. A Morse-Bott function is Morse-Bott-Smale if there exists a good gradient flow ϕt , i.e., adapted in
neighbourhoods of components of Cr(A), which satisfies the generalized Smale conditions.
Recall that we say p < q if there is a piecewise broken flow line from p to q. We have a similar definition for
F < F 0 . We also had the following :
Lemma 139. Assume that f is MBS11 with flow ϕt . Then F < F 0 implies λF < λF 0 and λ∗F > λ∗F 0 .
Example 140. Let π : X → B be a smooth fibre bundle. If f : B → R is Morse-Smale then f
generalized Morse-Smale.
11 We’ll
be abbreviating Morse-Bott-Smale by MBS from now on.
67
◦
π : X → R is
Let T be the graph of ϕt given by
T := {(x, ϕt (x) | x ∈ X, t ≥ 0} ⊂ X × X.
Consider the fibre product
UF ×F SF := {(x, y) ∈ UF ×F SF | πU (x) = πS (y)} ⊂ X × X.
Theorem 141. (J. Latschev)
Let f be a Morse-Bott function on X which satisfies the generalized Morse-Smale conditions for a good metric on X. Then
T and each fibre product UF ×F SF are manifolds of finite volume12 in X × X. Moreover, there exists an equation of
currents on X × X given by ∂T = ∆ − P where
X
P =
UF ×F SF
F
as F ranges over connected components of cr(f ).
Idea of the proof
We shall need the following :
Definition 142. A compact manifold with corners is a Hausdorff space M with an atlas of charts {(Uα , ψα )}α∈A
such that
homeo
ψα : Uα −−−→ (R+ × · · · × R+ ) ×Rn−m
{z
}
|
m times
such that coordinate changes are smooth.
The main dirty work is the following :
Theorem 143. (J. Latschev)
Assume MBS as before. The closure S F of the stable manifold SF of each connected critical manifold F is the image of a
smooth family
π̃S
SeF −→
F
of compact manifolds with corners under a smooth map
jF : SeF −→ X
whose restriction jF to Int(SeF ) is a diffeomorphism onto SF such that
jF
SeFO
?
Int(SeF )
O
F
∼
=
/S
OF
/ S? F
O
πF
π̃F
=
/F
Theorem 144. Under assumptions of the first theorem, there exists an operator equation
T◦d+d◦T=I−P
where
P(α) = lim ϕ∗t (α) =
t→∞
12 Everything
X
F
X
πS∗ F (πUF )∗ (α ∧ UF ) [SF ] =
ResF (α)[SF ].
F
is oriented here but I skipped it while writing it.
68
eF and taking (π̃U )∗ (j ∗ α).
Notice that (πU )∗ α really means taking jF∗ α on U
F
Proof
We need to prove the formula. There is a commutative diagram
pr2
eF ×F SeF
U
pr1
eF
U
/ Se
F
πS
/F
πU
of families of manifolds with corners which restricts to
pr2
UF ×F SF
pr1
UF
πU
/ SF
/F
πS
of manifolds of finite volume.
Let me remind you what the operator calculus gives you :
P(α) = (pr2 )∗ (pr1 )∗ αU
F ×F SF
We claim that
.
(pr2 )∗ (pr1 )∗ (α) = (πS )∗ (πU )∗ (α).
We have to prove this locally and it follows from
(x, u, v)
/ (x, v)
(x, u)
/ x.
This completes the proof.
Corollary 145. Suppose that for all F 0 < F
λF 0 + nF 0 + 1 < λF .
Then
H∗ (X) =
M
H∗−λF (F ).
F
Proof
Observe that
eF ×F SeF = (y, x) ∈ X × X | y > p > x .
UF ×F SF = image U
Fix F 0 < F and notice that
dim SF 0 = nF 0 + λF 0 < λF − 1
whence
nF 0 + λF 0 + λ∗F < λF + λ∗F − 1 ≤ n − 1.
The boundary of the closure of UF ×F SF consists of sets of the form
(y, x) ∈ UF × SF 0 | y > p > x
F0 < F
(y, x) ∈ UF 0 × SF | y > p > x
F < F 0.
69
Both of these sets have dimension less than n − 1. By the Federer support lemma
∂ [Uf ×F SF ] = 0.
P
Therefore, with P = F PF we see that PF is a chain map and
d◦T+T◦d=I−
X
PF
F
where PF : H∗ (X) → H∗ (SF ) ∼
= H∗−λF (F ) is induced by
PF : E l (X) −→ E 0l (X), α 7→ πS∗ ((πU )∗ α)[SF ].
Let ρ = (πU )∗ α. We need deg ρ ≤ dim F = nF if non-zero.
70
Lecture 23 - 3rd May ’11
We work over Z.
Fact 1
{UF } give a Whitney stratification of X, i.e.,
[
X=
Xk , X0 ⊂ X1 ⊂ · · · Xk ⊂ · · ·
k
such that Xk − Xk−1 is a manifold for any k.
Xk
Let Λ(Xk ) be the cone bundle over Xk . Each Xk − Xk−1 is a union of UF ’s. For example, if X = S 2
p
q
{p} ⊂ X with X − {p} = Uq .
∞
m
Definition 146. Consider
called totally transverse to a locally closed
a C -map of a simplex σ : ∆ → FX. This
submanifold U ⊂ X if σ ∆F is transverse to U for every face ∆ of ∆m .
The families transversality theorem implies that any finite set of maps σα : ∆mα → X can be made totally
transverse to a given finite set of locally closed submanifolds by a diffeomorphism of X arbitrarily close to the
identity map.
Definition 147. Let C∗t (X; Z) be the integral currents on X defined by smooth singular chains which are totally
transverse to {UF }.
Note that if σ ∈ C∗t (X; Z) then σ ∂∆ is automatically totally transverse. So there exists a natural mapping
t
t
∂ : Cm
(X; Z) −→ Cm−1
(X; Z).
There is also an isomorphism H∗t (X; Z) ∼
= H∗ (X; Z).
Fact 2 The domains of the operators T, P extend to C∗t (X; Z) and continue to satisfy d ◦ T + T ◦ d = I − P.
What is the image of P?
Recall that
jF
/S
SeF
F _
π

F
/X
71
where the fibres of π are manifold with corners. Define the stable bundle of σ to be
jF (fibre product) = jF (Seσ ) = Sσ
where
Seσ = {(x, y) ∈ ∆m × SeF | σ(x) = π(y)}.
Seσ
σ
jF
Sσ
F
Let SF be the integral currents defined by the stable bundles of smooth singular chains c =
σj : ∆m → F . Set
Sf = ⊕F SF .
P
nj σj where
We claim that the image of P is Sf . Keep in mind the Thom isomorphism
Hk (F ) −→ Hk+λF (SF ∪ ∞), σ 7→ SF (σ) = π −1 (σ) ∪ ∞.
Idea of the proof
We claim that on a simplex σ : ∆m → X we get
P(σ) = lim (ϕ−t )∗ (σ).
t→∞
This is true since we can think of σ as a generalized differential form σ
e of degree n − m, i.e.,
Z
Z
σ∗ β =
σ
e ∧ β, β ∈ E m (X).
∆m
X
Then we conclude that
P(e
σ ) = lim (ϕ∗t σ
e)
t→∞
and we’re done. Suppose now that σ t U = UF and
σ(∆m ) ∩ (U − U ) = φ.
σ
U
Notice that σ −1 (U ) = M is a compact submanifold of U . Since πU (M ) is a chain in F , if we write M =
where τj : ∆k → U then
X
(πU )∗ (M ) =
nj (πU ◦ τj ).
j
72
P
nj τj
Moreover, modulo stuff lower down we have the equality
P(σ) = lim ϕ−t (σ) = stable bundle of (πU )∗ (M ).
t→∞
The theorem implies that (Sf , ∂) is a subcomplex and H∗ (X; Z) ∼
= H∗ (Sf , ∂).
Given a stable chain SF (σ) the boundary
∂(SF (σ)) = lim ϕ−t (∂∆SF (σ)).
t→∞
This leads to a filtration of Sf
φ = F−1 ⊂ F0 ⊂ F1 ⊂ · · · ⊂ Fn = Sf
where
Fk :=
M
SF .
λF ≤k
Since F 0 < F implies that λF 0 ≤ λF we have ∂ : Fk → Fk whence these are subcomplexes. Therefore, there is a
r
spectral sequence (Ep,q
, dr ) such that
1
Ep,q
= Hp+q (Fp , Fp−1 ) =
M
λF =p
Thom
Hp+q (SF , SF − F ) ∼
=
M
Hq (F ).
λF =p
Spectral Sequences
We shall spend some time on this topic. Even if you’ve seen it before it never hurts to see it again.
Definition 148. A Z-bigraded module is a family E = {Ep,q }, p, q ∈ Z of modules over a ring R. A differential
d : E → E of bidegree (−r, r − 1) is a family of maps d : Ep,q → Ep−r,q+r−1 such that d2 = 0.
We can define
Hp,q (E) :=
ker dEp,q
d(Ep−r,q+r−1 )
If we define
.
En := ⊕p+q=n Ep,q
then d : En → En−1 .
Definition 149. A spectral sequence is a sequence E = (E r , dr ) of Z-bigraded differential modules where dr has
bidegree (−r, r − 1) together with isomorphisms
∼
=
E r+1 −→ H• (E r ).
Notice that E r determines E r+1 but not dr . That’s part of the definition. But as we’ll see, there is a canonical
choice!
73
Lecture 24 - 5th May ’11
Let me remind you of the definition of the spectral sequence.
Definition 150. A spectral sequence is a sequence E = (E r , dr ) of Z-bigraded differential modules where dr
has bidegree (−r, r − 1) together with isomorphisms
∼
=
E r+1 −→ H• (E r ).
Notice that E r determines E r+1 but not dr . There are times when it begins with r = 1 or with r = 2 and where
you begin matters!
Remark 151. There is a cohomology version where dr has bidegree (r, 1 − r). Moreover, we use Erp,q in this version.
r
Definition 152. E is a first quadrant spectral sequence if Ep,q
= 0 when p < 0 or q < 0 and for any r.
There is a picture of spectral sequences that’s good to keep in mind. Suppose we begin with (E 1 , d1 ). We get
B1 ⊂ B2 ⊂ B3 ⊂ · · · ⊂ Z 3 ⊂ Z 2 ⊂ Z 1 ⊂ E1
by setting
Z1
:= ker d1
B1
e2
Z
:= Im d1
e2
B
:= ker d2
:= Im d2
e2 and B
e 2 in Z 1 ⊂ E 1 under the map Z 1 → Z 1 /B 1 = E 2 .
and setting Z 2 and B 2 to be the pre-images of Z
Definition 153. The limit of the spectral sequence is given by the bigraded complex E ∞ := Z ∞ /B ∞ where
\
[
Z∞ =
Z r , B∞ =
Br .
r
If E is first quadrant then each
∞
Ep,q
r
is computed in finite time.
Suppose E is first quadrant. Then the bottom row contains no boundaries, i.e.,
p+1
1
2
∞
Ep,0
⊃ Ep,0
⊃ · · · ⊃ Ep,0
= Ep,0
.
Similarly, the first column entries are always cycles and therefore
1
E0,q
2
/ / E0,q
/ / ···
/ / E q+2 = E ∞ .
0,q
0,q
Definition 154. A morphism of spectral sequences f : E → E 0 is a family of homomorphisms f r : E r → E 0r of
bidegree (0, 0) such that for any r
f r ◦ dr = d0r ◦ f r .
0
Theorem 155. If f : E → E 0 is a morphism of spectral sequences and if f is an isomorphism for some r then f r is an
isomorphism for r0 ≥ r. Moreover, if E and E 0 are first quadrant then E ∞ ∼
= E 0∞ .
The proof is an easy exercise but it is an important fact.
We shall discuss the spectral sequence associated to a filtration.
Definition 156. A filtration of a module A is a sequence of submodules
· · · Fp−1 (A) ⊂ Fp (A) ⊂ Fp+1 (A) ⊂ · · · .
The associated graded module is
G(A) = ⊕p Gp (A), Gp (A) := Fp (A)/Fp−1 (A).
74
Definition 157. A homomorphism of filtered modules is a module homomorphism f : A → A0 such that
f (Fp (A)) ⊆ Fp (A0 ) for any p.
Suppose that (A, ∂) is a Z-graded differential module with |∂| = −1.
Definition 158. A filtration of a Z-graded differential module (A, ∂) will be a filtration by Z-graded differential
submodules.
It follows from this definition that if
· · · Fp−1 (A) ⊂ Fp (A) ⊂ Fp+1 (A) ⊂ · · ·
is a filtration of (A, ∂) then this induces a filtration of the Z-graded homology
· · · Fp−1 H• (A) ⊂ Fp H• (A) ⊂ Fp+1 H• (A) ⊂ · · ·
where Fp H• (A) is the image of H• (Fp (A)).
The Z-grading of A and the filtration {Fp (A)}p give us a filtration on An , i.e,
Fp (An ) := Fp (A) ∩ An
with ∂ : Fp (An ) → Fp (An−1 ). Therefore, Fp (An ) is a bigraded differential module. Write n = p + q where
p
is the filtration degree and q is the complementary degree and think of the previous module as Fp (Ap+q ) .
Definition 159. The filtration is bounded if for any n there exists α = α(n) and β = β(n) such that
0 = Fα (An ) ⊂ Fα+1 (An ) ⊂ · · · ⊂ Fβ (An ) = An .
Theorem 160. Each filtration Fp (A) of a Z-graded differential module A determines a spectral sequence (E r , dr ) r≥1 ,
as a covariant functor of (F, A), with natural morphisms
E 1 = H• Fp (A)/Fp−1 (A) .
If F is bounded then
E r =⇒ G H• (A) ,
∞
is computed in finite time and
i.e., Ep,q
∞ ∼
Ep,q
= Fp (Hp+q (A))/Fp−1 (Hp+q (A)).
I’m not going to do the proof in detail since you can read and these things become dry after a while. The idea is to
define approximate cycles of level r via
Zpr := {a ∈ Fp | ∂a ∈ Fp−r }.
r−1
Notice that Zp0 = Fp and set Bpr := ∂Zp+r−1
. This gives
Bp1 
e1 B
p

/ Bp2 
/ ··· 
/ Zp2 
/ Zp1 
/ Fp

/B
e2 / ··· 

/Z
e2 p
/ Ze1 
/ Fp /Fp−1 .
p
p
Example 161. The simplest example comes from a (finite) simplicial or CW complex X of dimension n. We have
S0 ⊂ S1 ⊂ · · · ⊂ Sn = X
where Sk is the k-skeleton of X. This induces a filtration of (C∗ (X), ∂) given by
Fp (C∗ (X)) := C∗ (Sp ) = C0 (X) ⊕ C1 (X) ⊕ · · · ⊕ Cp (X).
75
By definition
Ep1 = H∗ (Fp /Fp−1 ) ∼
= H∗ (Cp (X), ∂ ≡ 0) = Cp (X).
We can also think of Ep1 as the free abelian group generated by eα ’s, the p-cells in X. Therefore,
1
Ep,q
=
Cp (X) if q = 0
0 if q 6= 0.
We claim that d1 = ∂ : Cp (X) → Cp−1 (X). The short exact sequence
0 −→ (Fp , Fp−1 ) −→ (Fp+1 , Fp−1 ) −→ (Fp+1 , Fp ) −→ 0
induces a long exact sequence
/ H∗ (Fp+1 , Fp−1 )
H∗ (Fp , Fp−1 )
_??



??



?



?



δ ??


Ep1
H∗ (Fp+1 , Fp )
_??

??

??



d1 ??

1
Ep+1
This implies that (E 1 , d1 ) = (C∗ (X), ∂) and Ep2 = Hp (X) and after this dr = 0, i.e., the spectral sequence collapses
at r = 2.
76
Lecture 25 - 10th May ’11
Let me give you this picture we had last time. We have a filtration on A13
· · · ⊂ Fp−1 ⊂ Fp ⊂ Fp+1 ⊂ · · ·
where we set
Zpr
Bpr
:= {a ∈ Fp | ∂a ∈ Fp−r }
r−1
:= ∂Zp+r−1
.
We also have the filtration
Bp1 ⊂ Bp2 ⊂ · · · ⊂ Zp3 ⊂ Zp2 ⊂ Zp1 ⊂ A.
Example 162. (Homology)
Let X be a finite complex with Sk being its k-skeleton. We set A = C∗ (X) and Fp (A) = C∗ (Sp ). Then
M
Ep1 = H∗ (Fp , Fp−1 ) =
Zeα
p−cell eα
with d1 = ∂.
Example 163. (Cohomology)
Let S0 ⊂ S1 ⊂ · · · Sn = X be the same complex. If we want to do cohomology then set A = C ∗ (X) and let
F k (A) = ker{C ∗ (X) → C ∗ (Sk )}.
We have a filtration
F n ⊃ F n−1 ⊃ · · · ⊃ F 0 .
The cohomology spectral sequence tells us that E 2 = E ∞ = H ∗ (X) and
M
E1p = H ∗ (F p , F p−1 ) ∼
Ze∗α = C p (X)
=
p−cell eα
with d1 = δ.
Example 164. (K-Theory)
Let S0 ⊂ S1 ⊂ · · · Sn = X be the same complex. We set A = K(X) and let
F k (A) = ker{K(X) → K(Sk )}.
Notice that K i (pt) is Z for i even and is zero otherwise. The spectral sequence gives us
M
K(pt)e∗α = C p (X, K(pt)).
E1p = H∗ (F p , F p−1 ) =
α
Notice that E2p = H∗ (X, K(pt)) and d2 6= 0. In general, E ∞ =⇒ G(K(X)), the associated graded complex.
Example 165. (Fibre bundle)
π
Let F → X → B be a fibre bundle with X compact. Let
S0 ⊂ S1 ⊂ · · · ⊂ Sm = B
be the skeletal decomposition of B. Set A = C∗ (X) and
Fp (A) = C∗ (π −1 (Sp )).
13 I’ll
drop the A’s from here on when the context is clear.
77
We have the following :
1
Ep,q
= Hp+q (Fp , Fp−1 )
= Hp+q (π −1 (Sp ), π −1 (Sp−1 ))
_
= Hp+q
π −1 (epα ), ∂π −1 (epα )
=
with d1 = ∂. If we assume that B is simply connected then
2
Ep,q
= Hp (B, Hq (F )) =⇒ H∗ (X).
Example 166. (Sphere bundles)
π
2
Let F = S n−1 ,→ X → B be a sphere bundle with B simply connected. We know Ep,q
which exists possibly in
n
row 0 and row n − 1. The only differential of interest, i.e., possibly non-zero, is d of bidegree (−n, n − 1).
add spectral sequence picture
We get a long exact sequence
dn
dn−1
· · · −→ Hn+k (B) −→ Hk (B) −→ Hn+k−1 (X) −→ Hn+k−1 (B) −−−→ · · ·
which also goes by the name of Gysin sequence.
Example 167. We have a double complex
A∗,∗ , ∂ 0 , ∂ 00
such that
∂ 0 : Ap,q −→ Ap−1,q , ∂ 00 : Ap,q −→ Ap,q−1
and (∂ 0 )2 = (∂ 00 )2 = 0 and ∂ 0 and ∂ 00 commute with each other. There are two filtrations
M
M
Fq00 :=
Ar,s , Fq0 :=
Ar,s .
s≤q
r≤p
add picture of double complex
We have two spectral sequence such that
1 0
(Ep,q
)
= Hp,q (A∗,∗ , ∂ 0 )
1 00
(Ep,q
)
= Hp,q (A∗,∗ , ∂ 00 )
and both converge to H∗ (A∗,∗ , ∂ 0 + ∂ 00 ).
One of the main examples is the one which many of you already know. Let X be a complex manifold with
A = E p,q (X). We have
E10p,q = H p,q (A, ∂), E100p,q = H p,q (A, ∂)
0
and so E∞
converges to the de Rham cohomology of X, starting from the Dolbeault cohomology.
Example 168. (Kähler manifolds)
Let X be a compact Kähler manifold. Let V be a holomorphic vector field, i.e., LV (J) = 0. We start with the
usual double complex Ap,q = E p,n−q (X) with d0 = ∂ and d00 = ιV . Notice that
2
∂ = (ιV )2 = ∂ιV + ιV ∂ = 0.
The first spectral sequence has E1p,q = H p,n−q (X). Notice that the ιV is exact if V 6= 0, i.e., away from the zeroes
of V this does not have cohomology. For the second spectral sequence
E1p,q
= ιV -cohomology
= H q X, ker ιV p
Ω
= H q (X, S )
where S is supported in the zero set of V . It follows from
78
Theorem 169. If dim Z(V ) ≤ k then H l (X, S ) = 0 for any l > k if S is supported in Z(V ).
that if dim Z(V ) ≤ k then
H p,q (X) = 0 if |p − q| > k.
This is a theorem due to Carrell & Lieberman and Frankel. In particular, if V has isolated zeroes then the cohomology of X is of type (p, p).
1
Example 170. Let X be a S 1 -manifold and let V be a generating vector field. Let A = E ∗ (X)S be the complex of
S 1 -invariant differential forms. Consider d + ιV and notice that dιV + ιV d = mathcalLV = 0 on A. Then a little
bit of work shows that
1
E ∗ (X)S , d + ιV =⇒ HS∗ 1 (X).
Let us return to Morse theory. Consider a Morse function f : X → R with Smale condition for V , a
gradient like flow. Let S be the stable chains of F and set
M
Sf =
SF
F
We have seen that
H∗ (Sf , ∂) ∼
= H∗ (X; Z).
We have filtration given by Fp where
Fp :=
M
SF .
λ(F )≤p
We have seen before that there is a spectral sequence such that
E 1 p , q = Hp+q (Fp , Fp−1 ) =
M
Hq (F ).
λ(F )=p
This will lead us to Morse-Bott inequalities.
Let p, q ∈ Z[t]. By definition p > q if and only if p−q = (1+)Q(t) where Q(t) has non-negative coefficients.
Lemma 171. Let (C∗ , ∂) be a finite chain group with H∗ denoting its homology. Let
X
M (t) :=
(dim Ck )tk
k
p(t)
:=
X
(dim Hk )tk .
k
Then M > p.
Now look at the spectral sequence (E r , dr ) with E r+1 = H(E r ). Let
M r (t) =
X
(dim Ekr )tk .
k≥0
Then we conclude that
M 1 > M 2 > · · · > M ∞.
Now observe that
M 1 (t)
=
X X
dim Hq (F )tp+q
p,q λ(F )=p
=
tλ(F ) pF (t) = Hq (F )tq
79
Lecture 26 - 12th May ’11
I’m going to cover a lot of territory today.
Equivariant theory
Let X be a compact G-manifold and let f : X → R be G-invariant such that
(i) Cr(f ) consists of a finite number of G-orbits.
(ii) Hess f is non-degenerate on normals to orbits (Morse-Smale).
Suppose there exists a MBS flow ϕt and G ⊂ Um . We have
(EG)1 ⊂ (EG)2 ⊂ · · · , (EG)k = Stm (Cm+2k+2 ).
We want homology of XG := X × GEG. We do this by taking limits of X ×G (EG)k and I’m going to drop the
k’s from now on.The function f lifts to a function on X × EG and then descends to a function f˜ on X ×G EG.
Notice that
N
a
Cr(f ) =
Oj , Oj ∼
= G/Hj .
j=1
Therefore,
Cr(f˜)
=
N
a
(Oj × EG)/G
j=1
=
N
a
(G/Hj × EG)/G
j=1
=
N
a
(EG/Hj )
j=1
=
N
a
BHj .
j=1
We have a spectral sequence such that
1
Ep,q
=
M
HqG (Oj ) =
λj =p
M
Hq (BHj ) =⇒ H∗G (X).
λj =p
C∗ -actions on a Kähler manifold X
Let C∗ be the group under complex multiplication.
Average the Kahler form over S 1 and LV ω = 0 where V is the generating vector field,
0 = LV ω = d(ιV ω) + ιV (dω)
Lemma 172. (Frankel)
The following holds
ιV ω = df.
Notice that
Cr(f )
=
fixed point set of S 1 -action
=
finite union of compact complex submanifolds
a
Fj .
=
j
80
Theorem 173. (Sommese)
rieties and so is the closure of
The closures of the stable and unstable manifolds of each Fj are complex analytic subva{(z, ϕτ (z)) ∈ X × X | z ∈ X, τ ∈ C∗ }.
Corollary 174. ϕt , t ∈ R is a finite volume flow. Therefore,
d◦T+T◦d=I−P
Recall that
P(α) =
where Res α = πs∗ (πU )∗ αU and
X
Resj (α)[S(Fj )]
j
F
P
UF ?
??
?
πF ??
F
SF



  πS
As a special case let dim Fj = 0 for any j, i.e., isolated fixed points. Then H∗ (X; Z) is generated by
nj [Spj ] and
Z
X
P(α) =
nj [Spj ], nj =
α.
Uj
j
∗
∗
Corollary 175. If dim(X C ) = 0 then H∗ (X) is generated by the analytic subvarietiesP[Sp ], p ∈ X C . In fact, the
formula ∂T = ∆ − P is a homology in X × X between the diagonal ∆ and a cycle P = Up × Sp , a sum of complex
analytic Künneth components.
We actually have a rational equivalence of ∆ and P . We therefore have
C − cycles/rational equivalence ∼
= H∗ (X; Z).
∗
More generally, suppose that dimC (X C ) = k, i.e., the maximum dimension of the fixed point set is k.
Recall the formula for P(α). Let α be of type (r, s) and then
bidegree (πU )∗ α = (r − λ∗F , s − λ∗F ).
This is a form on F , a complex k-dimensional manifold. If (πU )∗ α 6= 0 then r − λ∗F ≤ k and s − λ∗F ≤ k. We can
conclude that
λ∗F
λ∗F
≤ r ≤
≤ s ≤
k + λ∗F
k + λ∗F .
We have this nice theorem which was first proved by fancy spectral sequence methods.
Theorem 176. (CL)
∗
If X is Kähler manifold with a C∗ -action such that dim(X C ) = 0 then
Hp,q (X) = 0 if |p − q| > k.
Novikov Theory
Let X be a compact manifold and consider ω ∈ E 1 (X) such that dω = 0. We assume that
(1) the zeroes of ω are all non-degenerate, i.e., ω = df locally and Hess f is non-degenerate.
(2) there exists a gradient like flow ϕt such that Sp t Uq for p, q ∈ Cr(ω).
e → X such that π ∗ ω = df for f : Y → R. There is a minimal such
First Idea There is a covering space π : X
1
space. Consider [ω] ∈ H (X; R) and let Γ be the kernel of the map
R
ω
π1 (X) −→ H1 (X; Z) −−→ R.
There is a covering space π : Y → X such that π1 (Y ) ∼
= Γ, whence [π ∗ ω] = 0 in H 1 (Y ). In other words,
∗
π ω = df .
81
Example 177. Let F : X → S 1 and ω = f ∗ (dθ). We have
/R
f
Y
π∗
X
π
/ S1.
F
picture of R-covering of a genus 2 surface by choosing one simple loop; unstable goes right; stable goes left;
f goes right
If x ∈ Cr(ω) then π −1 (x) = Zy such that π(y) = x.
Next Idea The stable and unstable manifolds Sy and Uy for y ∈ Cr(f ) are directional. Assume that the Deck
group Z is generated by g in the example before. Then f (gy) = f (y) + 1.
Definition 178. A closed set A ⊂ Y is compact/forward if
(1) A ∩ f −1 ([b, c]) is compact (or slab compact) for any b < c
(2) A ⊂ f −1 ([a, ∞)) for some a.
picture of infinite holed torus mapping down to R
One similarly has the notion of compact/backward and the intersection of these two types is compact. Each
unstable manifold is compact/forward and each stable manifold is compact/backward.
Definition 179. We define the following :
k
Ec↑
(Y )
= smooth k-forms with compact/forward support
k
Ec↓
(Y
0k
Ec↓ (Y
)
= smooth k-forms with compact/backward support
)
= currents of degree k with compact/backward support.
0k
Ec↑
(Y )
= currents of degree k with compact/forward support
There is a pairing
0(n−k)
k
Ec↑
(Y ) × Ec↓
(Y ) −→ R.
Example 180. Let M k ⊂ Y be an oriented k-dimensional compact/backward submanifold of locally finite volume,
0(n−k)
i.e., [M ] ∈ Ec↓
(Y ). Define
Z
k
[M ](ω) =
ω, ω ∈ Ec↑
(Y ).
M
Theorem 181. There is a gradient flow ϕt for ω on X such that on Y it is Morse-Smale. In particular, Sp , Up and T are
finite volume where
T = {(y, ϕt (y)) | t ∈ [0, ∞)} ⊂ X × X.
We also have ∂T = ∆ − P and we have
T
0∗−1
∗
: Ec↓
(Y ) −→ Ec↓
(Y )
∗
0∗
P, I : Ec↓
(Y ) −→ Ec↓
(Y ).
Morover, we can write
P(α) =
X
p∈cr(f )
Notice that S p is a compact/backward set.
82
Z
Up
α [Sp ].
0∗
Corollary 182. Let (S (f ), ∂) be the subcomplex of Ec↓
(Y ) of elements
s=
X
np [Sp ]
p∈F
where F is the compact/backward subset of cr(f ). We have the isomorphism
0∗
∗
H(Ec↓
(Y )) := Hc↓
(Y ) ∼
= H ∗ (Sc↓ (f ), ∂).
Novikov’s Idea Let t be the generator of Z, the Deck group going backward, i.e., f (ty) = f (y) + 1. Observe
∗
that t maps critical points to itself, Sp to itself and Up to itself. Moreover, t maps Ec↓
(Y ) to itself and for the other
0∗
three spaces as well. Not only this but powers of t act as well. If you think a little then Z[[t]] acts on Ec↓
(Y ) and
∗
−1
on Sc↓ (f ). Even more, Λ := Z[[t]][t ] acts on it. This is called the Novikov ring. We will be working with
ΛR = Λ ⊗Z R = R[[t]][t−1 ].
Lemma 183. The following holds :
(i) ΛR is a field.
(ii) Λ is a principal ideal domain.
That being said, go home and prove this for yourself.
Z
Proposition 184. Sc↓
(f ) is a finitely generated free module over Λ and the number of generators in degree p equals the
number of critical points of ω on X of index p.
p
R
(f ) is a finite dimensional vector space over ΛR and Sc↓
(f ) is of dimension the number critical points
Similarly, Sc↓
of index p.
This gives me the final theorem assuming all these conditions.
Theorem 185. There is an isomorphism of ΛR -vector spaces
∗
∗
Hc↓
(Y ) := H(Ec↓
(Y )) ∼
= H ∗ (Sc↓ (f )).
We finally cam write down the Morse-Novikov inequalties. Let
k
vk (Y ) := dimΛR Hc↓
(Y ).
P
P
Define P (t) =
vk (Y )tk and M (t) =
nλ tλ where nλ is the number of critical points of ω on X of index λ.
The inequality simply is
M > P.
83