0.10. Acyclic complexes and contractible manifolds 17

The full version can be found in (Spanier, 1966, page 397) or (Hatcher,

2002, page 371).

Theorem 0.8.4 (Relative Hurewicz Isomorphism). Let (X, A) be a (k −1)-

connected pair, k ≥ 2, such that A is nonempty and simply connected. Then

for each x0 ∈ A there is a natural isomorphism πk(X, A, x0) → Hk(X, A).

0.9. The Whitehead Theorem

The Whitehead Theorem allows one to detect algebraically that a map of

complexes is a homotopy equivalence.

Theorem 0.9.1 (Whitehead). A map f : K → L between simplicial com-

plexes (or CW complexes) is a homotopy equivalence if and only if f∗ :

πn(K) → πn(L) is an isomorphism for every n.

This particular statement of the theorem can be found in (Hatcher,

2002, page 346), for example. Here is a related result that is often easier

to apply, and which follows from the Whitehead Theorem, the Relative

Hurewicz Isomorphism Theorem, and a mapping cylinder construction.

Theorem 0.9.2. A map f : K → L between 1-connected simplicial com-

plexes (or CW complexes) is a homotopy equivalence if and only if f∗ :

H∗(K; Z) → H∗(L; Z) is an isomorphism.

0.10. Acyclic complexes and contractible manifolds

As an application of the theorems in the last few sections we briefly consider

acyclic and contractible spaces.

Definition. A space X is acyclic if

ˇ

H ∗(X; Z)

∼

=

0.

The following is an immediate consequence of the Hurewicz and White-

head Theorems.

Theorem 0.10.1. A 1-connected complex K is contractible if and only if it

is acyclic.

Example 0.10.2. There exists a compact 2-dimensional CW complex that

is acyclic but not contractible.

Proof. The classic example is the CW complex Y that has one 0-cell, two

1-cells a and b, and two 2-cells attached to the loops

a5b−3

and

b3(ab)−2.

The cellular chain complex for Y has the form · · · → 0 →

Z2

∂

− →

Z2

→ Z,

where ∂ is represented by the matrix A =

5 −3

−2 1

. Since det A = −1,

∂ is an isomorphism; hence H∗(Y ; Z)

∼

=

0 and Y is acyclic.