Hatcher 2.1.28

Hatcher, Algebraic Topology, Chapter 2, Section 1

28. Let X be the cone of the 1-skeleton of \Delta^3, the union of all line segments joining the points in the six edges of \Delta^3 to the barycenter of \Delta^3. Compute the local homology groups H_n(X,X-\left\{x\right\}) for all x\in X. Define \partial X to be the subspace of points x such that H_n(X,X-\left\{x\right\})=0 for all n, and compute the local homology groups H_n(\partial X,\partial X-\left\{x\right\}). 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.
Proof.

\square

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