WHAT IS K-HOMOLOGY ?
Paul Baum
Penn State
Texas A&M University
College Station, Texas, USA
April 2, 2014
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
1 / 56
Let X be a compact C ∞ manifold without boundary.
X is not required to be oriented.
X is not required to be even dimensional.
On X let
δ : C ∞ (X, E0 ) −→ C ∞ (X, E1 )
be an elliptic differential (or pseudo-differential) operator.
(S(T X ⊕ 1R ), Eσ ) ∈ K0 (·), and
Index(DEσ ) = Index(δ).
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
2 / 56
(S(T X ⊕ 1R ), Eσ )
w
w

Index(δ) = (ch(Eσ ) ∪ Td((S(T X ⊕ 1R )))[(S(T X ⊕ 1R )]
and this is the general Atiyah-Singer formula.
S(T X ⊕ 1R ) is the unit sphere bundle of T X ⊕ 1R .
S(T X ⊕ 1R ) is even dimensional and is — in a natural way — a Spinc
manifold.
Eσ is the C vector bundle on S(T X ⊕ 1R ) obtained by doing a clutching
construction using the symbol σ of δ.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
3 / 56
Minicourse of five lectures:
1. Dirac operatorX
2. Atiyah-Singer revisitedX
3. What is K-homology?
4. Beyond ellipticity
5. The Riemann-Roch theorem
The minicourse is based on joint work with Ron Douglas and is dedicated
to Ron Douglas.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
4 / 56
K-homology in algebraic geometry
Let X be a (possibly singular) projective algebraic variety / C.
Grothendieck defined two abelian groups:
0 (X) = Grothendieck group of algebraic vector bundles on X.
Kalg
K0alg (X) = Grothendieck group of coherent algebraic sheaves on X.
0 (X) = the algebraic geometry K-theory of X (contravariant).
Kalg
K0alg (X) = the algebraic geometry K-homology of X (covariant).
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
5 / 56
K-homology in topology
Problem
How can K-homology be taken from algebraic geometry to topology?
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
6 / 56
K-homology is the dual theory to K-theory. There are three ways in which
K-homology in topology has been defined:
Homotopy Theory K-theory is the cohomology theory and
K-homology is the homology theory determined by the Bott
(i.e. K-theory) spectrum.
This is the spectrum . . . , Z × BU, U, Z × BU, U, . . .
K-Cycles K-homology is the group of K-cycles.
C ∗ -algebras K-homology is the Atiyah-BDF-Kasparov
group KK ∗ (A, C).
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
7 / 56
Let X be a finite CW complex.
The three versions of K-homology are isomorphic.
Kjhomotopy (X)
homotopy theory
−→
Kj (X) −→ KK j (C(X), C)
←−
K-cycles
Atiyah-BDF-Kasparov
j = 0, 1
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
8 / 56
K-homology in topology
Problem
How can K-homology be taken from algebraic geometry to topology?
There are three ways in which this has been done:
Homotopy Theory K-homology is the homology theory
determined by the Bott spectrum.
K-Cycles K-homology is the group of K-cycles.
C ∗ -algebras K-homology is the Kasparov group KK ∗ (A, C).
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
9 / 56
Atiyah-BDF-Kasparov K-homology
M.F. Atiyah
Brown-Douglas-Fillmore
G.Kasparov
Let X be a finite CW complex.
C(X) = {α : X → C | α is continuous}
L(H) = {bounded operators T : H → H}
Any element in the Atiyah-BDF-Kasparov K-homology
group KK 0 (C(X), C)
is given by a 5-tuple (H0 , ψ0 , H1 , ψ1 , T ) such that :
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
10 / 56
H0 and H1 are separable Hilbert spaces.
ψ0 : C(X) −→ L(H0 ) and ψ1 : C(X) −→ L(H1 )
are unital ∗-homomorphisms.
T : H0 −→ H1 is a (bounded) Fredholm operator.
For every α ∈ C(X) the commutator T ◦ ψ0 (α) − ψ1 (α) ◦ T
∈ L(H0 , H1 ) is compact.
KK 0 (C(X), C) := {(H0 , ψ0 , H1 , ψ1 , T )}/ ∼
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
11 / 56
KK 0 (C(X), C) := {(H0 , ψ0 , H1 , ψ1 , T )}/ ∼
(H0 , ψ0 , H1 , ψ1 , T ) + (H00 , ψ00 , H10 , ψ10 , T 0 ) =
(H0 ⊕ H00 , ψ0 ⊕ ψ00 , H1 ⊕ H10 , ψ1 ⊕ ψ10 , T ⊕ T 0 )
−(H0 , ψ0 , H1 , ψ1 , T ) = (H1 , ψ1 , H0 , ψ0 , T ∗ )
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
12 / 56
Let X be a finite CW complex.
Any element in the Atiyah-BDF-Kasparov K-homology
group KK 1 (C(X), C)
is given by a 3-tuple (H, ψ, T ) such that :
H is a separable Hilbert space.
ψ : C(X) −→ L(H) is a unital ∗-homomorphism.
T : H −→ H is a (bounded) self-adjoint Fredholm operator.
For every α ∈ C(X) the commutator T ◦ ψ(α) − ψ(α) ◦ T ∈ L(H)
is compact.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
13 / 56
KK 1 (C(X), C) := {(H, ψ, T )}/ ∼
(H, ψ, T ) + (H0 , ψ 0 , T 0 ) = (H ⊕ H0 , ψ ⊕ ψ 0 , T ⊕ T 0 )
−(H, ψ, T ) = (H, ψ, −T )
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
14 / 56
Let X, Y be CW complexes and let f : X → Y be a continuous map.
Denote by f \ : C(X) ← C(Y ) the ∗-homomorphism
f \ (α) := α ◦ f
α ∈ C(Y )
Then f∗ : KK j (C(X), C) → KK j (C(Y ), C) is
f∗ (H, ψ, T ) := (H, ψ ◦ f \ , T )
j=1
f∗ (H0 , ψ0 , H1 , ψ1 , T ) := (H0 , ψ0 ◦ f \ , H1 , ψ1 ◦ f \ , T )
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
j=0
April 2, 2014
15 / 56
Cycles for K-homology
Let X be a CW complex.
Definition
A K-cycle on X is a triple (M, E, ϕ) such that :
1
M is a compact Spinc manifold without boundary.
2
E is a C vector bundle on M .
3
ϕ : M → X is a continuous map from M to X.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
16 / 56
Set K∗ (X) = {(M, E, ϕ)}/ ∼ where the equivalence relation ∼
is generated by the three elementary steps
Bordism
Direct sum - disjoint union
Vector bundle modification
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
17 / 56
Isomorphism (M, E, ϕ) is isomorphic to (M 0 , E 0 , ϕ0 ) iff ∃ a
diffeomorphism
ψ : M → M0
preserving the Spinc -structures on M, M 0 and with
ψ ∗ (E 0 ) ∼
=E
and commutativity in the diagram
MA
ψ
AA
AA
ϕ AAA
X
Paul Baum (Penn State)
/ M0
|
|
||
|| ϕ0
|
}
|
WHAT IS K-HOMOLOGY ?
April 2, 2014
18 / 56
Bordism (M0 , E0 , ϕ0 ) is bordant to (M1 , E1 , ϕ1 ) iff ∃ (Ω, E, ϕ) such
that:
1
2
3
Ω is a compact Spinc manifold with boundary.
E is C vector bundle on Ω.
(∂Ω, E|∂Ω , ϕ|∂Ω ) ∼
= (M0 , E0 , ϕ0 ) t (−M1 , E1 , ϕ1 )
−M1 is M1 with the Spinc structure reversed.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
19 / 56
(M0 , E0 , ϕ0 )
(−M1 , E1 , ϕ1 )
X
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
20 / 56
Direct sum - disjoint union
Let E, E 0 be two C vector bundles on M
(M, E, ϕ) t (M, E 0 , ϕ) ∼ (M, E ⊕ E 0 , ϕ)
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
21 / 56
Vector bundle modification
(M, E, ϕ)
Let F be a Spinc vector bundle on M
Assume that
dimR (Fp ) ≡ 0
mod 2
p∈M
for every fiber Fp of F
1R = M × R
S(F ⊕ 1R ) := unit sphere bundle of F ⊕ 1R
(M, E, ϕ) ∼ (S(F ⊕ 1R ), β ⊗ π ∗ E, ϕ ◦ π)
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
22 / 56
S(F ⊕ 1R )
π
M
This is a fibration with even-dimensional spheres as fibers.
F ⊕ 1R is a Spinc vector bundle on M with odd-dimensional fibers. Let Σ
be the spinor bundle for F ⊕ 1
Clif fC (Fp ⊕ R) ⊗ Σp → Σp
π ∗ Σ = β ∗ ⊕ β−
(M, E, ϕ) ∼ (S(F ⊕ 1R ), β ⊗ π ∗ E, ϕ ◦ π)
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
23 / 56
{(M, E, ϕ)}/ ∼= K0 (X) ⊕ K1 (X)
subgroup of {(M, E, ϕ)}/ ∼
consisting of all (M, E, ϕ) such that
Kj (X) =
every connected component of M
has dimension ≡ j mod 2, j = 0, 1
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
24 / 56
Addition in Kj (X) is disjoint union.
(M, E, ϕ) + (M 0 , E 0 , ϕ0 ) = (M t M 0 , E t E 0 , ϕ t ϕ0 )
Additive inverse of (M, E, ϕ) is obtained by reversing the Spinc structure
of M .
−(M, E, ϕ) = (−M, E, ϕ)
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
25 / 56
e ϕ)
DEFINITION. (M, E, ϕ) bounds ⇐⇒ ∃ (Ω, E,
e with :
1
2
3
4
Ω is a compact Spinc manifold with boundary.
e is a C vector bundle on Ω.
E
ϕ
e : Ω → X is a continuous map.
e ∂Ω , ϕ|
(∂Ω, E|
e ∂Ω ) ∼
= (M, E, ϕ)
REMARK. (M, E, ϕ) = 0 in K∗ (X) ⇐⇒ (M, E, ϕ) ∼ (M 0 , E 0 , ϕ0 ) where
(M 0 , E 0 , ϕ0 ) bounds.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
26 / 56
Let X, Y be CW complexes and let f : X → Y be a continuous map.
Then f∗ : Kj (X) → Kj (Y ) is
f∗ (M, E, ϕ) := (M, E, f ◦ ϕ)
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
27 / 56
Chern character in K-theory
X finite CW complex
ch : K j (X) −→
M
H j+2l (X; Q)
l
j = 0, 1
Q ⊗Z K j (X) −→
M
H j+2l (X; Q)
l
is an isomorphism of Q vector spaces.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
28 / 56
Chern character in K-homology
X finite CW complex
(M, E, ϕ) 7→ ϕ∗ (ch(E) ∪ T d(M ) ∩ [M ])
M
ch : Kj (X) −→
Hj+2l (X; Q)
l
j = 0, 1
Q ⊗Z Kj (X) −→
M
Hj+2l (X; Q)
l
is an isomorphism of Q vector spaces.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
29 / 56
Chern character and module structure
X a finite CW complex.
K ∗ (X) is a ring and K∗ (X) is a module over this ring.
Chern character respects the ring and module structure.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
30 / 56
Theorem (PB and R.Douglas and M.Taylor, PB and N. Higson and
T. Schick)
Let X be a finite CW complex.
Then for j = 0, 1 the natural map of abelian groups
Kj (X) → KK j (C(X), C)
is an isomorphism.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
31 / 56
For j = 0, 1 the natural map of abelian groups
Kj (X) → KK j (C(X), C)
is (M, E, ϕ) 7→ ϕ∗ [DE ]
where
1
2
3
DE is the Dirac operator of M tensored with E.
[DE ] ∈ KK j (C(M ), C) is the element in the
Atiyah-BDF-Kasparov K-homology of M determined by DE .
ϕ∗ : KK j (C(M ), C) → KK j (C(X), C) is the homomorphism of
abelian groups determined by ϕ : M → X.
Paul Baum (Penn State)
WHAT IS K-HOMOLOGY ?
April 2, 2014
32 / 56