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 ∗ (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.

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