2024-11-08

We have showed that

$$d(\mathcal{U},\mathcal{V})=\frac{1}{2}\inf \{\text{dis}(C)\mid C:\mathcal{U}\rightrightarrows \mathcal{V}\text{ is a correspondence}\}$$

where

$$\text{dis}(C)=\sup \{|d_{St}(\{U_i:i\in \sigma \})-d_{St}(\{V_i:i\in \tau \})|:(\sigma ,\tau )\in C\}$$

is a pseudometric. We seek an example where $d(\mathcal{U},\mathcal{V})=0$ but $\mathcal{U}$ and $\mathcal{V}$ are not isomorphic.

Let $p\in {\mathbb{R}}^2$ be a point and construct $\mathcal{U}=\{B_1(p)\}$ and $\mathcal{V}=\{B_{ϵ}(p){\}}_{ϵ\in [0,1]}$, where $B_{ϵ}(p)$ is the ball of radius $ϵ$ centered at $p$. Clearly the Mapper complexes corresponding to $\mathcal{U}$ and $\mathcal{V}$ are not isomorphic. Consider the multi-valued map $C_{ϵ}:\mathcal{U}\rightrightarrows \mathcal{V}$ given by

$$C_{ϵ}=\{(\{p\},B_{{ϵ}^{\prime }}(p)):{ϵ}^{\prime }\ge \sqrt{1-ϵ}\}.$$

Then under the Lebesgue measure,

$$\begin{array}{rl}\text{dis}(C_{ϵ})& =\underset{\alpha ,\beta \in (0,1)}{\sup }\left\{\left|\frac{\mu (B_1(p))}{\mu (B_1(p))}-\frac{\mu (B_{\alpha }(p))}{\mu (B_{\beta }(p))}\right|:\beta \ge \alpha \ge \sqrt{1-ϵ}\right\}\\ & =\underset{\alpha ,\beta \in (0,1)}{\sup }\left\{\left|1-\frac{{\alpha }^2}{{\beta }^2}\right|:\beta \ge \alpha \ge \sqrt{1-ϵ}\right\}\\ & =1-\frac{1-ϵ}{1}\\ & =ϵ.\end{array}$$

Therefore,

$$d(\mathcal{U},\mathcal{V})\le \frac{1}{2}\inf ϵ=0.$$

Unfortunately, $C_{ϵ}$ is not correspondence. While $C_{ϵ}$ is a multi-valued map, $C_{ϵ}^{\top }$ is not. In fact the only correspondence here is $C=\mathcal{U}\times \mathcal{V}$. Under this correspondence,

$$\text{dis}(C)=\underset{\alpha ,\beta \in (0,1)}{\sup }\left\{\left|\frac{\mu (B_1(p))}{\mu (B_1(p))}-\frac{\mu (B_{\alpha }(p))}{\mu (B_{\beta }(p))}\right|:\beta \ge \alpha >0\right\}=1-\frac{0}{1}=1.$$

Moreover, the cover $\mathcal{V}$ is not a valid Mapper cover.

Let ${\text{Nrv}}_{St}(\mathcal{U})$ and ${\text{Nrv}}_{St}(\mathcal{V})$ be non-isomorphic Mapper complexes and assume that $d(\mathcal{U},\mathcal{V})=0$. We have that $H_{\ast }({\text{Nrv}}_{St}(\mathcal{U}))$ and $H_{\ast }({\text{Nrv}}_{St}(\mathcal{V}))$ are 0-interleaving, i.e., isomorphic, but this does not imply that the complexes themselves are isomorphic. For example, $S^1\times S^1$ and $S^1\vee S^1\vee S^2$ have the same homology groups but are not isomorphic.