Hatcher-Algebraic-Topology · Problem 1.A.5

The Fundamental Group

Mathematics — Graph Theory · graphs-and-free-groups

Problem

Construct a connected graph

X

and maps

f, g : X \to X

such that

fg = \mathbb{1}

but

f

and

g

do not induce isomorphisms on

\pi_1

. [Note that

f_* g_* = \mathbb{1}

implies that

f_*

is surjective and

g_*

is injective.]