Algebraic Topology · Problem 10

Homotopy Theory — Allen Hatcher

Mathematics — Topology · cw-complex-s1-sn-cell-attachment

Problem

Let the CW complex

X

be obtained from

S^1 \vee S^n

n \geq 2

, by attaching a cell

e^{n+1}

by a map representing the polynomial

p(t) \in \mathbb{Z}[t, t^{-1}] \approx \pi_n(S^1 \vee S^n)

, so

\pi_n(X) \approx \mathbb{Z}[t, t^{-1}]/p(t)

. Show

\pi'_n(X)

is cyclic and compute its order in terms of

p(t)

. Give examples showing that the group

\pi_n(X)

can be finitely generated or not, independently of whether

\pi'_n(X)

is finite or infinite.