Algebraic Topology · Problem 11

Homotopy Theory — Allen Hatcher

Mathematics — Topology · pi2-relative-1skeleton

Problem

Let XX be a connected CW complex with 1-skeleton X1X^1. Show that π2(X,X1)≈π2(X)×K\pi_2(X, X^1) \approx \pi_2(X) \times K where KK is the kernel of π1(X1)→π1(X)\pi_1(X^1) \to \pi_1(X), a free group. Show also that the map π2′(X)→π2′(X,X1)\pi'_2(X) \to \pi'_2(X, X^1) need not be injective by considering the case X=RP2X = \mathbb{R}P^2 with its standard CW structure.

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