Algebraic Topology · Problem 2C.4

Cohomology — Allen Hatcher

Mathematics — Topology · lefschetz-number-simplicial-homeomorphism

Problem

X

is a finite simplicial complex and

f: X \to X

is a simplicial homeomorphism, show that the Lefschetz number

\tau(f)

equals the Euler characteristic of the set of fixed points of

f

. In particular,

\tau(f)

is the number of fixed points if the fixed points are isolated. [Hint: Barycentrically subdivide

X

to make the fixed point set a subcomplex.]