19 December 2003, University of Paris 13.
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
JIE WU
Department of Mathematics
National University of Singapore
Singapore 117543
Republic of Singapore
matwuj@nus.edu.sg
http://www.math.nus.edu.sg/∼matwujie
This talk will consist of:
1. The Exponent Problem.
2. The Cohen Groups.
3. Shuffle Relations.
4. Configuration Spaces
†Supported in part by the Academic Research Fund of the National University of Singapore.
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2
JIE WU
1. The Exponent Problem
It is not clear at least to me whether the answer to the Barratt
conjecture is yes or no, but I will talk some progress along this
direction. These progress seem interesting by themselves.
Barratt Conjecture. Let X be a path-connected co-Hspace such that the degree map [pr ] : X → X is null homotopic.
Then pr+1π∗(X) = 0.
Barratt Conjecture for Maps. Let X be a path-connected
co-H-space and let f : X → Y be a map such that [f ] is of
order pr in [X, Y ]. Then pr+1 Im(f∗ : π∗(X) → π∗(Y )) = 0.
The first statement follows from the second statement because
one can choose Y = X and f = idX .
So far essentially only knew that the Barratt conjecture holds
for the cases when X is a mod pr Moore space for p > 2. For
p = 2 and r > 1, it was known by F. Cohen that π∗(P n(2r ))
has a bounded exponent. For mod 2 Moore spaces, it is even
not clear where π∗(P n(2)) has a bounded exponent. It was
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
3
known that Z/8 occurs as a summand of π∗(P n(2)) for infinite
times in different dimensions for n ≥ 3.
• For technical reasons, we assume that X is a double (or triple
or even higher) suspension in case.
• From the cofibre sequence
Sn
[pr ]
-
Sn
j
-
P n+1(pr ),
the Barratt conjecture (for double suspensions) is equivalent to
that
pr+1 : Im((idY ∧j)∗ : π∗(Y ∧ S n) → π∗(Y ∧ P n+1(pr ))
for n ≥ 2 and any Y because a f : Y ∧ S n → Z is of order pr
in [Y ∧ S n, Z] if and only if it factors through Y ∧ P n+1(pr ),
and idY ∧j : Y ∧ S n → Y ∧ P n+1(pr ) is of order pr .
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JIE WU
F Basic Ideas (due to F. Cohen): Assume that X = ΣX 0 is
a suspension. Given a map f : ΣX 0 → Y , let f 0 : X 0 → ΩY be
the adjoint map of f . Observe that Ωf : ΩX = ΩΣ0X → ΩY
is the (unique up to homotopy) H-map such that Ωf |X 0 ' f 0.
• Step 1. Construct certain group H (we will go through how
to construct H) with a group homomorphism
θ : H → [ΩΣX 0, ΩY ].
The group H is noncommutative and the image of θ contains
[Ωf ] and many other homotopy classes given by composite of
Hopf invariants and Whitehead products. Roughly speaking
H should be big enough such that we can do something for
making relations between the powers of [Ωf ] and other types
of maps.
• Step 2. Consider the commutative diagram
H
θ
-
[ΩΣX 0, ΩY ]
Ω : [g] 7→ [Ωg]
?
H̃
θ̃-
2
?
0
[Ω ΣX , Ω2Y ],
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
5
where H̃ is certain extension of certain quotient of H by adding
possible relations (for instance since [Ω2Σ0X, Ω2Y ] is abelian,
the composite Ω ◦ θ factors through the abelianization of H)
and, if possible, adding new elements.
Assume that the answer to the Barratt conjecture would be
positive. We wish that H̃ has a bounded exponent by adding
enough relations to H. (Certainly one could consider triple
loop spaces if there are good relations occur after looping twice
or more.)
• I have no ideas what H̃ looks like. But I will talk certain
relations to H. These relations do not seem good enough for
answering the Barratt conjecture, but they are canonical in
some sense.
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JIE WU
2. The Cohen Groups
Let X be a path-connected space. Recall that the James
construction J(X) ' ΩΣX. From the James filtration
Xn
qn
?
?
X = J1(X) ⊆ · · · ⊆Jn(X)⊆ · · · ⊆ J(X),
there is a tower of groups
[J(X), ΩY ] · · · [Jn(X), ΩY ] · · · [X, ΩY ]
∩
qn∗
n
?
[X , ΩY ]
by suspension splitting theorem, that is,
[J(X), ΩY ] = lim[Jn(X), ΩY ]
n
with representation
qn∗ : [Jn(X), ΩY ] ⊂
-
[X n, ΩY ].
The coordinate inclusions and projections
di : X n+1 → X n
di : X n → X n+1
(x0 , x1 , . . . , xn ) 7→ (x0 , . . . , xi−1 , xi+1 , . . . , xn )
(x0 , x1 , . . . , xn−1 ) 7→ (x0 , . . . , xi−1 , ∗, xi , . . . , xn−1 )
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
7
induce faces di = di∗ and cofaces di = d∗i on G = {[X n+1, ΩY ]}
such that G is a ∆-group and a co-∆-group with the relation


di−1 dj for j < i,









i
dj d =
id
for j = i,









 i
d dj−1 for j > i.
This relation is different from the third identity of simplicial
sets, and we call this kind of objects bi-∆-groups. (I do not
know any references for studying these objects.)
• The faces are used by obtaining [Jn(X), ΩY ] by the following
lemma.
F Lemma. Let X and Z be path-connected spaces. Suppose
that X is a co-H-space or Z is an H-space. Then [Jn+1(X), Z]
is the equalizer of the faces di = di∗ : [X n+1, Z] → [X n, Z] for
0 ≤ i ≤ n.
• The cofaces can be used as operations for constructing Hopf
invariants combinatorially. It turns out that many classical
results such as the distributivity law hold for any bi-∆-group.
• There are two ways to obtain the Cohen group H. The first
way was given by F. Cohen. Given a map f : X → ΩY , he
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JIE WU
constructed a group homomorphism from the free group
θ : Fn+1 → [X n+1, ΩY ]
by sending the generators to
X n+1
coordinate projections
-
X
f
-
ΩY.
When X is a suspension, he discovered certain commutator
relations (the iterated commutators on generators such that
one of the generators occurs at least twice) and so the map
θ factors through a quotient group Kn+1 of Fn+1. Then he
took the equalizer of faces to obtain a group Hn with a group
homomorphism
θ : Hn → [Jn(X), ΩY ]
for any suspension X and any given map f : X → ΩY . By
taking limits, then he obtained a group homomorphism
θ : H = lim Hn → [J(X), ΩY ]
n
for any suspension X and any given map f : X → ΩY .
One important property of Hn is that:
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
9
FCohen Theorem. H is a progroup and the kernel of Hn Hn−1 is isomorphic to Lie(n), where Lie(n) over a ring R (in
this case R = Z) is given as follows:
Let V be a free n-dimensional R-module with a basis {x1, . . . , xn}.
Then Lie(n) is the R-submodule of V ⊗n spanned by Lie elements
[[xσ(1), xσ(2)], . . . , xσ(n)]
for σ ∈ Sn. The symmetric group Sn acts on Lie(n) by permuting letters {x1, . . . , xn}.
• The elements in H are given by (possibly infinite) products
of composites
J(X)
Hn
-
J(X (n))
J(α)
-
J(X (n))
ΩWn
-
J(X)
f˜
-
ΩY,
where Hn is the James-Hopf map, Wn is the Whitehead product, f˜ is the H-map such that f˜|X ' f and α is a linear
combination of permutations.
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JIE WU
3. Shuffle Relations
• Observe that Ω : [X, Y ] → [ΩX, ΩY ] = [ΣΩX, Y ] is induced
by the evaluation map σ : ΣΩX → X. Recall that there is a
homotopy pull-back diagram
σ
ΣΩX
-
X
∆
?
?
X ∨X
⊂
-
X ×X
¯:X →
for any space X. In particular, the reduced diagonal ∆
X ∧ X is null homotopic after looping.
The shuffle relations are essentially obtained by considering
elements in H that occur as composite
J(X)
¯
∆
-
J(X) ∧ J(X)
any map
-
ΩY.
¯ ' ∗.
These maps are null homotopic after looping because Ω∆
¯ : J(X) → J(X) ∧
F Lemma. If X is a co-H-space, then ∆
J(X) preserves the filtration up to homotopy, where the filtration on J(X) ∧ J(X) is given by the product filtration.
The proof is almost straightforward. Let µ0 : X → X ∨ X be
the comultiplication. Then the diagonal ∆ : X n → X n ×X n =
X 2n preserves the filtration up to homotopy, namely (up to
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
11
homotopy) ∆ maps into the subspace of X 2n that has at most
n coordinates different from the basepoint. By taking quotients,
one get that (up to homotopy) ∆ : Jn(X) → Jn(X) × Jn(X)
S
maps into the subspace
Ji(X) × Jj (X), and so does the
i+j≤n
reduced diagonal.
• Let ψ̄ : J(X) → J(X) ∧ J(X) be the filtered map with ψ̄ '
¯ Then there is a commutative diagram of cofibre sequences
∆.
Jn−1(X) ⊂
-
ψ̄
Jn(X)
-
ψ̄
X (n)
shuffles ψ̄
?
?
[
Ji(X) ∧ Jj (X)
n−1
_
?
⊂ -
i+j≤n−1
[
Ji(X) ∧ Jj (X)
i+j≤n
-
X (i) ∧ X (n−i)
i=1
F Lemma. Let X be a path-connected co-H-space. Then
the composite
Jn(X) → X (n)
n−1
_
ψ̄
-
X (i) ∧ X (n−i)
any map
-
ΩY
i=1
is null homotopic after looping.
• Let R be the quotient group of H by adding the shuffle relations. Then the composite H → [J(X), ΩY ] → [ΩJ(X), Ω2Y ]
factors through R. The group R has the following properties:
• R is abelian.
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JIE WU
• There is commutative diagram of the tower of groups
H · · · Hn
?
?
R · · · Rn
Hn−1 · · · H1
?
?
Rn−1 · · · R1
Theorem. Let Kern be the kernel of Rn → Rn−1. Then there
is an exact sequence of groups
0 → Ext−1
R(Sn ) (Lie(n), Lie(n)) → Kern → EndR(Sn ) (Lie(n)),
where the ground ring R = Z, Z(p) (if X is a p-local space) or
Z/pr (if f : X → ΩY is of order pr ).
• The composition operation on [J(X), J(X)] induces an operation on H such that H is a semi-ring and θ : H → [J(X), J(X)]
corresponding the f = idΣX is the morphism of semi-rings. The
abelian group R has the better property that R is a ring and
the quotient H → R is a morphism of semi-rings.
• Paul Selick and I have been to introduce a functor Amin(X)
which roughly speaking is the functorial smallest retract of
J(X) that contains the bottom cell. There is a subring Rmin of
R such that Rmin is a local ring with a representation Rmin →
[ΩAmin(X), ΩAmin(X)].
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
13
4. Configuration Spaces
Now we are going to give some remarks on using configuration
spaces for understanding the evaluation map σ : ΣΩ2Σ2X →
ΩΣ2X. Let (M, M0) be a pair of manifolds. Recall that the
configuration space
C(M, M0; X) =
∞
a
F (M, n) ×Sn X n/ ≈
n=1
filtered by configuration length with
Dn(M, M0; X) = Cn/Cn−1 = F (M, n)/F (M |M0, n)∧Sn X (n),
where F (M |M0, n) is the subspace of F (M, n) consisting of all
configurations with at least one coordinate in M0.
We use the following basic property of configuration spaces:
F Let M be a smooth compact manifold and let M0 and N
be the smooth compact submanifolds of M with codimN = 0.
If N/M0 ∩ N or X is path connected, then
C(N, N ∩ M0; X) → C(M, M0; X) → C(M, N ∪ M0; X)
is a quasifibration.
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JIE WU
• Let E be the pull-back of the diagram
- C(M × (I, I0 ∪ I1 ); X)
E
q
?
?
C(M × (I, I0 ∪ I0.5 ); X) ∨ C(M × (I, I0.5 ∪ I1 ); X) ⊂ - C(M × (I, I0 ∪ I0.5 ∪ I1 ); X),
where I = [0, 1] and It is a small neighborhood of t for t ∈ I.
Since q is a quasi-fibration and q is homotopic to the diagonal
map, the space
E ' ΣΩC(M × (I, I0 ∪ I1); X) ' ΣC(M × I; X)
and the map
σ : E → C(M × (I, I0 ∪ I1); X)
is an evaluation map. It follows that the evaluation map
σ : ΣC(M × I; X) → C(M × (I, ∂I); X) ' C(M ; ΣX)
preserves that configuration filtration and so
ΣCn−1(M × I; X)
-
ΣCn(M × I; X)
σn−1
ΣDn(M × I; X)
σ̄n
σn
?
Cn−1(M ; ΣX)
-
?
-
Cn(M ; ΣX)
?
-
Dn(M ; ΣX)
THE EXPONENT PROBLEM IN HOMOTOPY THEORY
15
F Theorem. The map
σ̄n : ΣDn(M × I; X)
-
Dn(M ; ΣX)
is homotopic to the suspension of the quotient map
F (M × I; n)+ ∧Sn X (n)
-
F (M × I; n)/An ∧Sn X (n),
where An is the subspace of F (M × I; n) consisting of all configurations with the property that at least one coordinate lies
in M × I0 and at least one coordinate lies in M × I1.
• Roughly speaking the Sn-equivariant map
(F (M × I, n), ∅)
-
(F (M × I, n), An)
gives the evaluation map in certain sense.
• For the case M = I, the composite
ΣDn(R2; X)
σ̄n
-
(ΣX)(n)
n−1
_
ψ̄
(ΣX)(i) ∧ (ΣX)(n−i)
-
i=1
is null homotopic.
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JIE WU
• For the Barratt conjecture, one might consider
ΣCn(R2; X)
-
ΣDn(R2; X)
σ̄n
σn
?
?
Jn(ΣX)
?
cofibre
-
certain maps
(ΣX)
(n)
-
Σ2Cn−1(R2; X)
certain maps
certain maps
?
ΩY
w
w
w
w
w
w
w
w
w
w
- ΩY
-