Algebraic Topology · Problem 2.2.42

Homology — Allen Hatcher

Mathematics — Topology · homology-with-coefficients

Problem

Let

X

be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that

H_1(X; \mathbb{Z})

is free abelian of rank

n > 1

, so the group of automorphisms of

H_1(X; \mathbb{Z})

GL_n(\mathbb{Z})

, the group of invertible

n \times n

matrices with integer entries whose inverse matrix also has integer entries. Show that if

G

is a finite group of homeomorphisms of

X

, then the homomorphism

G \to GL_n(\mathbb{Z})

assigning to

g: X \to X

the induced homomorphism

g_*: H_1(X; \mathbb{Z}) \to H_1(X; \mathbb{Z})

is injective. Show the same result holds if the coefficient group

\mathbb{Z}

is replaced by

\mathbb{Z}_m

with

m > 2

. What goes wrong when

m = 2