Algebraic Topology · Problem 23

Homotopy Theory — Allen Hatcher

Mathematics — Topology · hurewicz-extension-surjectivity

Problem

Extend the Hurewicz theorem by showing that if

X

is an

(n-1)

-connected CW complex, then the Hurewicz homomorphism

h: \pi_{n+1}(X) \to H_{n+1}(X)

is surjective when

n > 1

, and when

n = 1

show there is an isomorphism

H_2(X)/h(\pi_2(X)) \approx H_2\bigl(K(\pi_1(X), 1)\bigr)

. [Build a

K(\pi_n(X), n)

from

X

by attaching cells of dimension

n+2

and greater, and then consider the homology sequence of the pair

(Y, X)

where

Y

X

with the

(n+2)

-cells of

K(\pi_n(X), n)

attached. Note that the image of the boundary map

H_{n+2}(Y, X) \to H_{n+1}(X)

coincides with the image of

h

, and

H_{n+1}(Y) \approx H_{n+1}\bigl(K(\pi_n(X), n)\bigr)

. The previous exercise is needed for the case

n > 1