Hatcher-Algebraic-Topology · Problem 7

The Fundamental Group

Mathematics — Graph Theory · homotopy-stationary-boundary

Problem

Define

f : S^1 \times I \to S^1 \times I

f(\theta, s) = (\theta + 2\pi s, s)

, so

f

restricts to the identity on the two boundary circles of

S^1 \times I

. Show that

f

is homotopic to the identity by a homotopy

f_t

that is stationary on one of the boundary circles, but not by any homotopy

f_t

that is stationary on both boundary circles. [Consider what

f

does to the path

s \mapsto (\theta_0, s)

for fixed

\theta_0 \in S^1