Hatcher-Algebraic-Topology · Problem 11

Homotopy Theory

Mathematics — Graph Theory · relative-pi2-cw-complex

Problem

Let

X

be a connected CW complex with

1

-skeleton

X^1

. Show that

\pi_2(X, X^1) \approx \pi_2(X) \times K

where

K

is the kernel of

\pi_1(X^1) \to \pi_1(X)

, a free group. Show also that the map

\pi'_2(X) \to \pi'_2(X, X^1)

need not be injective by considering the case

X = \mathbb{R}P^2

with its standard CW structure.