Hatcher-Algebraic-Topology · Problem 20

The Fundamental Group

Mathematics — Graph Theory · center-fundamental-group-homotopy

Problem

Suppose

f_t : X \to X

is a homotopy such that

f_0

and

f_1

are each the identity map. Use Lemma 1.19 to show that for any

x_0 \in X

, the loop

f_t(x_0)

represents an element of the center of

\pi_1(X, x_0)

. [One can interpret the result as saying that a loop represents an element of the center of

\pi_1(X)

if it extends to a loop of maps

X \to X