Hurewicz theorem explained

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

Statement of the theorems

The Hurewicz theorems are a key link between homotopy groups and homology groups.

Absolute version

For any path-connected space X and positive integer n there exists a group homomorphism

h_*\colon\pi_n(X)\toH_n(X),

called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator

u_n\in

	n)
H
	n(S

, then a homotopy class of maps

f\in\pi_n(X)

is taken to

f_*(u_n)\inH_n(X)

The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.

n\ge2

, if X is

(n-1)

-connected (that is:

\pi_i(X)=0

for all

i<n

), then

\tilde{H_i}(X)=0

for all

i<n

, and the Hurewicz map

h_*\colon\pi_n(X)\toH_n(X)

is an isomorphism. This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map

h_*\colon\pi_n+1(X)\toH_n+1(X)

is an epimorphism in this case.

n=1

, the Hurewicz homomorphism induces an isomorphism

\tilde{h}_*\colon\pi_1(X)/[\pi_1(X),\pi_1(X)]\toH_1(X)

, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version

(X,A)

and integer

k>1

there exists a homomorphism

h_*\colon\pi_k(X,A)\toH_k(X,A)

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both

and

are connected and the pair is

(n-1)

-connected then

H_k(X,A)=0

for

k<n

and

H_n(X,A)

is obtained from

\pi_n(X,A)

by factoring out the action of

\pi_1(A)

. This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by as a statement about the morphism

\pi_n(X,A)\to\pi_n(X\cupCA),

where

denotes the cone of

. This statement is a special case of a homotopical excision theorem, involving induced modules for

n>2

(crossed modules if

n=2

), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

Triadic version

For any triad of spaces

(X;A,B)

(i.e., a space X and subspaces A, B) and integer

k>2

there exists a homomorphism

h_*\colon\pi_k(X;A,B)\toH_k(X;A,B)

from triad homotopy groups to triad homology groups. Note that

H_k(X;A,B)\congH_k(X\cup(C(A\cupB))).

The Triadic Hurewicz Theorem states that if X, A, B, and

C=A\capB

are connected, the pairs

(A,C)

and

(B,C)

are

(p-1)

-connected and

(q-1)

-connected, respectively, and the triad

(X;A,B)

(p+q-2)

-connected, then

H_k(X;A,B)=0

for

k<p+q-2

and

H_p+q-1(X;A)

is obtained from

\pi_p+q-1(X;A,B)

by factoring out the action of

\pi_1(A\capB)

and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental

\operatorname{cat}ⁿ

-group of an n-cube of spaces.

Simplicial set version

The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.^[1]

Rational Hurewicz theorem

Rational Hurewicz theorem: Let X be a simply connected topological space with

\pi_{i(X) ⊗}\Q=0

for

i\leqr

. Then the Hurewicz map

h ⊗ \Q\colon\pi_{i(X) ⊗}\Q\longrightarrowH_i(X;\Q)

induces an isomorphism for

1\leqi\leq2r

and a surjection for

i=2r+1

Notes and References

, III.3.6, 3.7