An Approximate Nerve Theorem

Abstract

The nerve theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an $\varepsilon$-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an $\varepsilon$-acyclic cover of the space. Our approximations are stated in terms of interleaving distance between persistence modules. Using the Mayer–Vietoris spectral sequence, we prove upper bounds on the interleaving distance between the persistence module of the underlying space and the persistence module of the nerve of the cover. To prove the best possible bound, we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.

Appendix A: Convergence of the Mayer–Vietoris Spectral Sequence

The aim of this appendix is to briefly describe the basic idea of the proof of Theorem 2.30. Let $(M,\partial ^0,\partial ^1)$ be the double complex associated with a filtered cover of a filtered simplicial complex, i.e., a pair $(X,{\mathcal {U}})$, and $(E^r,d^r)$ the spectral sequence associated with this double complex, as defined in Sect. 2.3. As mentioned there, the spectral sequence associated with a double complex $(M,\partial ^0,\partial ^1)$ is just a tool to compute the homology of the associated total complex $({\mathrm {Tot}}(M),D)$, namely $(E^r,d^r)$ will converge to ${\mathsf {H}}_*({\mathrm {Tot}}(M))$. This standard fact can be established by a series of elementary but tedious computations, so we do not replicate the proof here, see, for instance, [31]. To prove Theorem 2.30, it is therefore sufficient to show that the homology of the total complex ${\mathrm {Tot}}(M)$ is isomorphic to the homology of $(X,f)$.

In fact, the double complex $(M,\partial ^0,\partial ^1)$ has a geometric counterpart, namely the filtered Mayer–Vietoris blowup complex $X_{\mathcal {U}}$ associated with $(X,{\mathcal {U}})$. The total complex ${\mathrm {Tot}}(M)$ arises as the chain complex associated with $X_{\mathcal {U}}$ and M arises from a filtration on ${\mathrm {Tot}}(M)$ which in turn is induced by a natural filtration of $X_{\mathcal {U}}$.

So far, we have been working mostly with filtered simplicial complexes; however, in the case at hand, it is slightly more convenient to work with filtered CW complexes and cellular homology. Any filtered simplicial complex $X$ gives rise to a filtered CW complex $X^{\mathrm {CW}}$ whose cellular homology is isomorphic to the simplicial homology of $X$. In fact, the corresponding chain complexes are isomorphic. Each simplex $\sigma$ in $X$ is assigned a cell $e_{\sigma }$ in $X^{\mathrm {CW}}$. The cartesian product $X_1^{\mathrm {CW}}\times X_2^{\mathrm {CW}}$ of two such complexes again has the structure of a CW complex whose cells are given as $e_{\sigma _1}\times e_{\sigma _2}$ for each pair of simplices $\sigma _1$ in $X_1$ and $\sigma _2$ in $X_2$. The blowup complex is a subcomplex of such a product.

Definition A.1

The filtered Mayer–Vietoris blowup complex associated with $(X,{\mathcal {U}})$ is the filtered CW complex $(X_{\mathcal {U}},{\mathcal {F}}_{\mathcal {U}})$, where $X_{\mathcal {U}}\le X\times {\mathcal {N}}$ is given by

$$\begin{aligned} X_{\mathcal {U}}=\bigcup _{\sigma \in U_I}e_\sigma \times e_I \end{aligned}$$

and the filtration ${\mathcal {F}}_{\mathcal {U}}=(X_{\mathcal {U}}^j)_{j\in {\mathbb {Z}}}$ is given by

$$\begin{aligned} X_{\mathcal {U}}^j=\bigcup _{\sigma \in U_I^j}e_\sigma \times e_I. \end{aligned}$$

Let $({\mathsf {C}}^{X}_*,\partial ^{X})$ be the (persistent) cellular chain complex associated with $X^{\mathrm {CW}}$ and let $({\mathsf {C}}^{{\mathcal {N}}}_*,\partial ^{{\mathcal {N}}})$ be the cellular chain complex associated with ${\mathcal {N}}^{\mathrm {CW}}$. Let $({\mathsf {C}}_*,\partial )$ be the cellular chain complex associated with the blowup complex $X_{\mathcal {U}}$. Explicitly, each ${\mathsf {C}}_n$ is the free ${\mathbb {k}}[t]$-module, generated by the graded set of all cells $e_\sigma \times e_I$ with $\dim \sigma +\dim I=n$, where the grading is given by $\deg (e_\sigma \times e_I)$, i.e., the birth times of the cells in the filtration of the blowup complex. Since the blowup complex is a subcomplex of $X^{\mathrm {CW}}\times {\mathcal {N}}^{\mathrm {CW}}$, the boundary homomorphisms $\partial _n$ are simply restrictions of the boundary homomorphisms of the chain complex associated with this product. These satisfy the following relation:

$$\begin{aligned} \partial _n(e_\sigma \times e_I)=\partial ^0_n(e_\sigma ) \times e_I+(-1)^{\dim \sigma }e_\sigma \times \partial ^1_n(e_I). \end{aligned}$$

Taking into account the isomorphisms between $({\mathsf {C}}^{X}_*,\partial ^{X})$ and $({\mathsf {C}}^{{\mathcal {N}}}_*,\partial ^{{\mathcal {N}}})$ and the corresponding simplicial chain complexes, it follows that $({\mathsf {C}}_*,\partial )\cong ({\mathrm {Tot}}_*(M),D)$. For comparison, here is the boundary formula for the latter chain complex, written out in full. If $(\sigma ,I)$ is a pair with $\dim \sigma =q$ and $\dim I=p$ such that $p+q=n$, we have

$$\begin{aligned} D_n(\sigma ,I)= & {} \sum _{k=0}^q(-1)^k t^{\deg (\sigma ,I)-\deg (\sigma _k,I)}(\sigma _k,I)\nonumber \\&+\,(-1)^q\sum _{l=0}^p(-1)^l t^{\deg (\sigma ,I)-\deg (\sigma ,I_l)} (\sigma ,I_l). \end{aligned}$$

This also explains the grading from which the double complex structure of M arises. Namely for each pair p, q with $p+q=n$, let $N_{p,q}\le {\mathsf {C}}_n$ be the submodule freely generated by all cells $e_\sigma \times e_I$ with $\dim \sigma =q$ and $\dim I=p$. Then, we have ${\mathsf {C}}_n=\bigoplus _{p+q=n}N_{p,q}$ and $\partial ^{X}\times {\text {id}}$ and ${\text {id}}\times \partial ^{{\mathcal {N}}}$ respect this grading. The aforementioned isomorphism of $({\mathsf {C}}_*,\partial )\cong ({\mathrm {Tot}}_*(M),D)$ isomorphically maps the double complex structure of N into that of M. Therefore, the homology of the total complex $({\mathrm {Tot}}(M),D)$ is precisely the (persistent) cellular homology of the blowup complex. In other words, we have:

Proposition A.2

The homology of the total complex is isomorphic to the homology of the filtered blowup complex:

$$\begin{aligned} {\mathsf {H}}_*({\mathrm {Tot}}(M),D)\cong {\mathsf {H}}^{\mathrm {CW}}_*(X_{\mathcal {U}},{\mathcal {F}}_{\mathcal {U}}). \end{aligned}$$

It only remains to check that ${\mathsf {H}}^{\mathrm {CW}}_*(X_{\mathcal {U}},{\mathcal {F}}_{\mathcal {U}})\cong {\mathsf {H}}_*(X,{\mathcal {F}})$. To see this, it suffices to construct a homotopy equivalence of the two spaces, in the filtered sense. Let $\pi :X_{\mathcal {U}}\rightarrow X^{\mathrm {CW}}$ be the natural projection (to the first component) and let $\pi ^j:X_{\mathcal {U}}^j\rightarrow (X^j)^{\mathrm {CW}}$ be the appropriate restriction. It is a standard fact [24, Proposition 4G.2] that these projections are homotopy equivalences. Finally, these maps obviously also respect the filtration, i.e., for each $j_1\le j_2$, the diagram

commutes, because the projections simply forget the second component, whereas the information from the first component remains unchanged. Therefore, as claimed in Theorem 2.30, the Mayer–Vietoris spectral sequence of $(X,{\mathcal {U}})$ converges to the cellular persistent homology of $X^{\mathrm {CW}}$ and therefore to the simplicial persistent homology of $X$.