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 \(f: X \to X\).
Solution
Let \(G\) be the \(1\)-skeleton of \(\Delta^3\), so \(G\cong K_4\), and write \[ X=CG \] with cone point \(c\). The graph \(G\) has \(4\) vertices and \(6\) edges, hence \[ \widetilde H_1(G)\cong\mathbb Z^{6-4+1}=\mathbb Z^3. \]
1. At the cone point \(c\),\[ H_n(X,X-\{c\})\cong \begin{cases} \mathbb Z^3,&n=2,\\ 0,&n\ne2. \end{cases} \]
Since \(X\) is contractible and \(X-\{c\}\) deformation retracts onto \(G\), the long exact sequence of the pair gives \[ H_n(X,X-\{c\})\cong\widetilde H_{n-1}(G). \] The stated groups follow from the homology of \(K_4\).
2. If \(x\) lies in the interior of one of the four radial edges from \(c\) to a vertex of \(G\), then\[ H_n(X,X-\{x\})\cong \begin{cases} \mathbb Z^2,&n=2,\\ 0,&n\ne2. \end{cases} \]
Three triangular \(2\)-cells meet along each such radial edge. A neighborhood of \(x\) is homeomorphic to \[ \mathbb R\times C(P_3), \] where \(P_3\) is a discrete set of three points and \(x\) corresponds to \((0,\text{cone point})\). Local homology is shifted by the open \(\mathbb R\) factor, so \[ H_n(X,X-\{x\})\cong\widetilde H_{n-2}(P_3). \] Since \(\widetilde H_0(P_3)\cong\mathbb Z^2\), the claim follows.
3. If \(x\) lies in the interior of one of the six triangular \(2\)-cells, then\[ H_n(X,X-\{x\})\cong \begin{cases} \mathbb Z,&n=2,\\ 0,&n\ne2. \end{cases} \]
Such a point has a neighborhood homeomorphic to \(\mathbb R^2\), whose local homology is \(\mathbb Z\) in degree \(2\) and zero otherwise.
4. If \(x\in G\), then\[ H_n(X,X-\{x\})=0 \qquad\text{for all }n. \]
At an interior point of an edge of \(G\), a neighborhood in \(X\) is a half-plane, whose local homology at a boundary point vanishes. At a vertex of \(G\), the local link in \(X\) is a tree with one central vertex and three leaves, hence contractible; local homology is the reduced homology of this link shifted by one, so it vanishes in every degree.
Thus \[ \boxed{\partial X=G.} \]
5. The local homology in \(\partial X=G\) is\[ H_n(G,G-\{x\})\cong \begin{cases} \mathbb Z,&n=1,\quad x\text{ in the interior of an edge},\\ \mathbb Z^2,&n=1,\quad x\text{ a vertex},\\ 0,&\text{otherwise}. \end{cases} \]
At an interior edge point, \(G\) is locally a line, giving local \(H_1\cong\mathbb Z\). At a vertex, three half-edges meet. The local link is three points, so \[ H_1(G,G-\{x\})\cong\widetilde H_0(P_3)\cong\mathbb Z^2. \] All other local groups vanish.
6. The homeomorphism group of \(X\) has exactly five point-orbits:\[ \{c\}, \quad \text{open radial edges}, \quad \text{open $2$-cells}, \quad \text{vertices of }G, \quad \text{open edges of }G. \]
The local homology calculations distinguish these five types: the first three by ranks \(3,2,1\) of local \(H_2\), and the last two by ranks \(2,1\) of local \(H_1\) inside the intrinsically defined subspace \(\partial X\).
Conversely, the tetrahedral symmetry group is transitive on the vertices and edges of \(G\), hence on the corresponding radial edges and triangular \(2\)-cells. Within any one open cell, a homeomorphism supported in that cell (and extended over its star when necessary) can move any chosen interior point to any other. Thus each listed stratum is a single orbit.
7. A subset \(A\subseteq X\) satisfies\[ f(A)\subseteq A \qquad\text{for every homeomorphism }f:X\to X \]if and only if it is a union of these five orbits.
Applying the hypothesis to \(f^{-1}\) gives \(A\subseteq f(A)\), so in fact \(f(A)=A\) for every homeomorphism. Therefore \(A\) must be a union of point-orbits. Conversely every union of orbits is invariant. By 6 the five displayed strata are exactly the orbits.
Hence there are exactly \[ \boxed{2^5=32} \] such invariant subsets.