Algebraic Topology · Problem 3H.5

Homotopy Theory — Allen Hatcher

Mathematics — Topology · cohomology-with-local-coefficients-graphs

Problem

X

is a finite connected graph with

\pi_1(X)

free on

g > 0

generators, show that

H^n(X; \mathbb{Z}[\pi_1 X])

is zero unless

n = 1

, when it is

\mathbb{Z}

when

g = 1

and the direct sum of a countably infinite number of

\mathbb{Z}

's when

g > 1

. [Use Proposition 3H.5 and compute

H^n_c(\tilde{X})

\varinjlim H^n(\tilde{X}, \tilde{X} - T_i)

for a suitable sequence of finite subtrees

T_1 \subset T_2 \subset \cdots

\tilde{X}

with

\bigcup_i T_i = \tilde{X}