Math 527 - Homotopy Theory
Spring 2013
Homework 11 Solutions
Problem 1. Show that a path-connected space is weakly equivalent to a product of EilenbergMacLane spaces if and only if it admits a Postnikov tower of principal fibrations with trivial
k-invariants (all of them).
Note. Here, we follow Hatcher’s convention that the k-invariants are used to build the Postnikov tower of X starting from P1 X and not P0 X. In other words, by “Postnikov tower of
principal fibrations”, we mean that the maps Pn X → Pn−1 X are principal fibrations for all
n ≥ 2. Using n ≥ 1 instead would force π1 X to be abelian.
Solution. Some preliminary observations.
1. A path-connected space X is weaklyQequivalent to a product of Eilenberg-MacLane spaces
if and only if it is weakly equivalent to i≥1 K(πi X, i).
2. A projection πB : B × F B is always a fibration, so that the sequence
π
ι
B
F →
− B × F −→
B
is a fiber sequence. Here ι = (cb0 , idF ) : F → B × F denotes the “slice inclusion” ι(f ) = (b0 , f ).
3. The homotopy fiber of the constant map c : X → Y is
F (c) = {(x, γ) ∈ X × Y I | γ(0) = c(x), γ(1) = y0 }
= {(x, γ) ∈ X × Y I | γ(0) = γ(1) = y0 }
= X × ΩY.
Iterating the homotopy fiber once more yields the fiber sequence
π
ι
c
X
ΩY →
− X × ΩY −→
X→
− Y.
In particular, the fibration πB : B × F B can be extended to the right by the constant map
if and only if F admits a delooping.
Now onto the proof of the statement.
∼
(⇒) Assume given a (zigzag of, but WLOG a single) weak equivalence ϕ : X −
→
1
Q
i≥1
K(πi X, i).
Then the successive projections
..
.
Qn
i=1
C
K(πi X, i) = Pn X
Qn−1
i=1
?
K(πi X, i) = Pn−1 X
..
.
K(π1 X, 1) × K(π2 X, 2) = P2 X
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ϕ
X
∼
/
Q
i≥1
/
K(πi X, i)
*
K(π1 X, 1) = P1 X
∗ = P0 X
form a Postnikov tower for X.
The truncation map between successive stages Pn X → Pn−1 X is a projection with fiber
K(πn X, n), which admits a delooping K(πn X, n + 1) since πn X is abelian (as n ≥ 2).
By observation (3), Pn X → Pn−1 X is a principal fibration which can be extended to the right
by the constant map
∗
Pn X → Pn−1 X →
− K(πn X, n + 1)
so that the k-invariant kn−1 ∈ H n+1 (Pn−1 X; πn X) is trivial.
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(⇐) Assume all k-invariants of X are trivial, i.e. for all n ≥ 2 we have fiber sequences
∗
Pn X → Pn−1 X →
− K(πn X, n + 1).
By observation (3), this implies the equivalence
Pn X ' Pn−1 X × ΩK(πn X, n + 1)
' Pn−1 X × K(πn X, n).
Repeating this equivalence inductively, we conclude that for all n ≥ 1, the Postnikov stages of
X are products
n
Y
Pn X '
K(πi X, i).
i=1
Since these truncation maps Pn X → Pn−1 X are projections, in particular fibrations, the homotopy limit of the tower is (equivalent to) its strict limit. We conclude:
∼
X−
→ holim Pn X
n
' holim
n
' lim
n
∼
=
∞
Y
n
Y
K(πi X, i)
i=1
n
Y
K(πi X, i)
i=1
K(πi X, i)
i=1
and thus X is weakly equivalent to a product of Eilenberg-MacLane spaces.
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Problem 2. Let X be a path-connected CW complex and G a group. Show that the map
π1 : [X, K(G, 1)]∗ → HomGp (π1 (X), G)
is a bijection.
Solution. WLOG X has a single 0-cell. Indeed, X is pointed homotopy equivalent to such a
CW complex, and the functors on both sides [−, K(G, 1)]∗ and HomGp (π1 (−), G) are invariant
under pointed homotopy equivalence.
WLOG X is 2-dimensional. Indeed, the skeletal inclusion ι2 : X2 ,→ X induces an isomorphism
'
ι2∗ : π1 (X2 ) −
→ π1 (X) and a bijection
'
→ [X2 , K(G, 1)]∗
ι∗2 : [X, K(G, 1)]∗ −
as shown in the notes from 5/29.
True for wedges of circles. When X is a wedge of circles X '
a bijection, as shown by the commutative diagram:
W
[ j S 1 , K(G, 1)]∗
π1
W
j∈J
S 1 , then π1 does induce
W
HomGp π1 ( j S 1 ), G
/
∼
=
∼
=
HomGp (∗j π1 (S 1 ), G)
Q
j [S
1
, K(G, 1)]∗
Q
j
π1
/
Q
j
HomGp (π1 (S 1 ), G)
Q
jψ
Q
j
π1 K(G, 1)
∼
=
'
/
Q
j
∼
=
G
'
where ψ : π1 K(G, 1) −
→ G is some fixed identification.
True in general. Let X be a 2-dimensional CW complex with a single 0-cell. WLOG all
attaching maps of the 2-cells are pointed, so that X = X2 sits in a cofiber sequence
_
_
S 1 → X 1 → X2 →
S 2.
(1)
By the theorem on the fundamental group of CW complexes, applying π1 to this specific cofiber
sequence (1) yields a right exact sequence of groups
_
π1 ( S 1 ) → π1 (X1 ) π1 (X2 ) → 0.
Applying HomGp (−, G) then yields a left exact sequence of pointed sets, which is the bottom
row in the diagram below.
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Applying [−, K(G, 1)]∗ to the cofiber sequence (1) yields an exact sequence of pointed sets. The
natural transformation π1 yields a map of exact sequences:
[
W
S 2 , K(G, 1)]∗ = 0
/ [X2 , K(G, 1)]∗
/ [X1 , K(G, 1)]∗
π1
0
π1
∼
=
/ HomGp (π1 (X1 ), G)
/ HomGp (π1 (X2 ), G)
/ [W S 1 , K(G, 1)]∗
π1
∼
=
/ HomGp π1 (W S 1 ), G .
W
Because X1 ' S 1 is also a wedge of circles, the two downward maps to the right are bijections,
and hence so is the downward map
π1 : [X2 , K(G, 1)]∗ → HomGp (π1 (X2 ), G) .
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Problem 3. Let X be a CW complex, with n-skeleton Xn , and let Y be a path-connected
simple space. Let n ≥ 2, and let fn , gn : Xn → Y be two maps which agree on Xn−1 , i.e.
fn |Xn−1 = gn |Xn−1 .
Let d(fn , gn ) ∈ C n (X; πn Y ) denote their difference cochain.
Show that fn ' gn rel Xn−2 holds if and only if [d(fn , gn )] = 0 ∈ H n (X; πn Y ) holds, i.e.
d(fn , gn ) is a coboundary.
Solution. Since Y is path-connected and simple, we can safely ignore basepoints and work
with unpointed maps.
WLOG X = Xn .
Consider the map
S : (Xn × ∂I) ∪ (Xn−1 × I) → Y
defined by
S|Xn ×{0} = fn
S|Xn ×{1} = gn
S|Xn−1 ×{t} = fn−1 for all t ∈ I.
(The letter S was chosen for “Stationary”.)
The condition fn ' gn rel Xn−2 can be stated as being able to extend the restriction
Sn−1 := S|Xn ×∂I∪Xn−2 ×I
to all of Xn × I. In other words, S = Sn is defined on the relative n-skeleton of the relative
CW complex
(Xn × I, Xn × ∂I)
and we want to extend its restriction Sn−1 from the relative (n − 1)-skeleton to the relative
(n + 1)-skeleton Xn × I. There exists such an extension if and only if the obstruction
class of Sn
c(Sn ) ∈ C n+1 (Xn × I, Xn × ∂I; πn Y )
is a coboundary.
The short exact sequence of cellular chain complexes
0 → C∗ (X × ∂I) → C∗ (X × I) → C∗ (X × I, X × ∂I) → 0
yields a short exact sequence of cellular cochain complexes
0 → C ∗ (X × I, X × ∂I; πn Y ) → C ∗ (X × I; πn Y ) → C ∗ (X × ∂I; πn Y ) → 0.
Using the fact that C∗ (I) is finitely generated and free in each degree, we obtain the isomorphism
C n+1 (Xn × I, Xn × ∂I; πn Y ) ∼
= C n (Xn ; πn Y ) ⊗Z C 1 (I)
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(2)
and moreover, the coboundary operator in the relative cellular cochain complex C ∗ (Xn ×I, Xn ×
∂I; πn Y ) corresponds to the coboundary in C ∗ (Xn ; πn Y ). In other words, the diagram
C n+1 (Xn × I, Xn × ∂I; πn Y ) o
∼
=
O
∼
=
C n (Xn ; πn Y ) ⊗Z C 1 (I)
/
O
C n (Xn ; πn Y )
O
δ⊗id
δ
C n (Xn × I, Xn × ∂I; πn Y ) o
∼
=
δ
∼
=
C n−1 (Xn ; πn Y ) ⊗Z C 1 (I)
/
C n−1 (Xn ; πn Y )
commutes.
Therefore, the obstruction class c(Sn ) ∈ C n+1 (Xn × I, Xn × ∂I; πn Y ) is a coboundary
if and only if the corresponding cochain in C n (Xn ; πn Y ) is a coboundary.
Relative (n + 1)-cells of (Xn × I, Xn × ∂I) are of the form enα × e1 for some n-cell enα of Xn with
attaching map ϕα : S n−1 → Xn−1 and characteristic map
Φα : (Dn , S n−1 ) → (Xn , Xn−1 ).
Here e1 denotes the unique 1-cell of the interval I.
The value of the cochain c(Sn ) on the relative (n + 1)-cell enα × e1 is the composite
/
∂(Dn × D1 )
∂Dn × D1 ∪ Dn × ∂D1
/
Xn−1
ϕα ×idI ∪Φα ×id∂I
S
(Xn × I)n
8
/
Y.
× I ∪ Xn × ∂I
By definition of S, that composite is homotopic to the map d(fn , gn )(enα × e1 ) ∈ πn Y (or minus
it, depending on our sign convention in the definition of the difference construction). This
proves the equality
c(Sn ) = ±d(fn , gn )
via the isomorphism (2).
Therefore the obstruction class c(Sn ) is a coboundary if and only if the difference
cochain d(fn , gn ) is a coboundary.
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