2024-11-15

After meeting with Bala, he suggested I look at the following spaces to prove that $d(\mathcal{U},\mathcal{V})=0$ does not imply isomorphic: $S^1\vee S^1\bigsqcup S^1\vee S^1$ and $S^1\bigsqcup S^1\vee S^1\vee S^1$. These two spaces are not isomorphic but have the same homology. Let $\mathcal{U}$ be the balls of radius $r$, where $1/2<r<\sqrt{3}/3$, centered at

$$X_{\mathcal{U}}=\left\{(0,0),\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),(1,0),\left(\frac{3}{2},\frac{\sqrt{3}}{2}\right),(2,0),(4,0),\left(\frac{9}{2},\frac{\sqrt{3}}{2}\right),(5,0),\left(\frac{11}{2},\frac{\sqrt{3}}{2}\right),(6,0)\right\}.$$

Let $\mathcal{V}$ be balls of radius $r$ centered at

$$X_{\mathcal{V}}=\left\{(0,0),\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),(1,0),(3,0),\left(\frac{7}{2},\frac{\sqrt{3}}{2}\right),(4,0),\left(\frac{9}{2},\frac{\sqrt{3}}{2}\right),(5,0),\left(\frac{11}{2},\frac{\sqrt{3}}{2}\right),(6,0)\right\}.$$

There are no 2-simplices. All 1-simplices are born at approximately 0.987 (when $r\approx 0.54$).

The complex ${\text{Nrv}}_{St}(\mathcal{U})$ have the following 1-simplices:

$$\begin{array}{r}[1,2],[1,3],[2,3],[3,4],[3,5],[4,5]\\ [6,7],[6,8],[7,8],[8,9],[8,10],[9,10]\end{array}$$

where the numbers are the indices of each ball. The complex ${\text{Nrv}}_{St}(\mathcal{V})$ have the following 1-simplices:

$$\begin{array}{r}[1,2],[1,3],[2,3]\\ [4,5],[4,6],[5,6],[6,7],[6,8],[7,8],[8,9],[8,10],[9,10]\end{array}$$

Define the correspondence $C:\mathcal{U}\rightrightarrows \mathcal{V}$ given by

$$C(i)=\{\begin{array}{ll}5& i=4\\ 4& i=5\\ i& \text{otherwise}\end{array}$$

where $i$ is the index of the ball.

Then the 1-simplex $[i,j]\in {\text{Nrv}}_{St}(\mathcal{U})$ is mapped to the 1-simplex $[i,j]$ if $i,j\notin \{4,5\}$ and the remaining 1-simplices have maps $[4,5]\mapsto [4,5]$, $[4,6]\mapsto [5,6]$, and $[5,6]\mapsto [4,6]$. But all 1-simplices are born at the same time in both complexes, so

$$\text{dis}(C)=0.$$