16 0. Prequel

For years the classic monograph by W. Hurewicz and H. Wallman (1948)

stood as the standard dimension theory reference. The more recent book by

J. van Mill (1989) is an excellent alternative.

0.8. The Hurewicz Isomorphism Theorem and its

localization

The Hurewicz Theorem relates homotopy and homology groups. The fol-

lowing statement appears on page 394 of Spanier (1966) and on page 366 of

Hatcher (2002).

Theorem 0.8.1 (Hurewicz Isomorphism). Let X be a (k − 1)-connected

space, k ≥ 2, with x0 ∈ X. Then there is a natural isomorphism πk(X, x0) →

Hk(X).

Corollary 0.8.2. If X is 1-connected and Hi(X)

∼

= 0 for 1 ≤ i ≤ k, then

πi(X, x0)

∼

=

0 for i ≤ k.

There is also a useful local version of the theorem that does not appear

in any of the standard references on algebraic topology.

Theorem 0.8.3 (Local Hurewicz). Suppose V ⊂ U0 ⊂ · · · ⊂ Uk, k ≥ 2, are

open sets such that Hk(V ) → Hk(U0) is trivial and πq(Uq) → πq(Uq+1) is

trivial for 0 ≤ q ≤ k − 1. Then πk(V ) → πk(Uk) is trivial.

Proof. Consider any map α :

Sk

→ V . As [α] = 0 in Hk(U0), there exist a

subdivision L of

Sk

and a singular (k + 1)-chain c = Σjnjσj carried by U0

such that Σiα#(τi) = ∂c, where {τi} denotes a collection of 1–1 simplicial

maps

∆k

→ L, one for each k-simplex of L, determined by some ordering

of the vertices. Let K denote a geometric realization of the finite, singular

complex determined by the {σj}; here K contains L as a subcomplex, and

α : L → V ⊂ U0 has a natural extension β : |K| → U0. Let K be the

union of K and the cone on its (k − 1)-skeleton. Since πq(Uq) → πq(Uq+1)

is trivial, we can extend β over successive skeleta to a map β : |K | → Uk.

Now

[Sk]

is zero in Hk(K) and hence in Hk(K ). One can easily check that

K is (k − 1)-connected. By the Hurewicz Isomorphism Theorem,

[Sk]

= 0

in πk(K ). Application of β confirms that [α] = 0 in πk(Uk).

Theorem 0.8.3 is also known as the Eventual Hurewicz Theorem—see

(Ferry, 1979, Proposition 3.1) and (Quinn, 1979, Theorem 5.2).

Several applications require a relative version of the Hurewicz Theorem.

A complete statement of the relative Hurewicz Theorem must take account

of the action of π1 on the higher homotopy groups. In order to avoid that

technicality we state the relative theorem only in the simply connected case.