Hatcher-Algebraic-Topology · Problem 2.1.28

Homology

Mathematics — Graph Theory · local-homology

Problem

Let

X

be the cone on the 1-skeleton of

\Delta^3

, the union of all line segments joining points in the six edges of

\Delta^3

to the barycenter of

\Delta^3

. Compute the local homology groups

H_n(X, X - \{x\})

for all

x \in X

. Define

\partial X

to be the subspace of points

x

such that

H_n(X, X - \{x\}) = 0

for all

n

, and compute the local homology groups

H_n(\partial X, \partial X - \{x\})

. Use these calculations to determine which subsets

A \subset X

have the property that

f(A) \subset A

for all homeomorphisms

f: X \to X