nerve complex

Let $I$ be a set of indices and $C$ be a family of sets $(U\_i)\_\{i\backslash in\; I\}$. The nerve of $C$ is a set of finite subsets of the index set $I$. It contains all finite subsets $J\backslash subseteq\; I$ such that the intersection of the $U\_i$ whose subindices are in $J$ is non-empty:{{Cite Matousek 2007}}, Section 4.3{{Rp|page=81|location=}}

:$N(C)\; :=\; \backslash bigg\backslash \{J\backslash subseteq\; I:\; \backslash bigcap\_\{j\backslash in\; J\}U\_j\; \backslash neq\; \backslash varnothing,\; J\; \backslash text\{\; finite\; set\}\; \backslash bigg\backslash \}.$

In Alexandrov's original definition, the sets $(U\_i)\_\{i\backslash in\; I\}$ are open subsets of some topological space $X$.

The set $N(C)$ may contain singletons (elements $i\; \backslash in\; I$ such that $U\_i$ is non-empty), pairs (pairs of elements $i,j\; \backslash in\; I$ such that $U\_i\; \backslash cap\; U\_j\; \backslash neq\; \backslash emptyset$), triplets, and so on. If $J\; \backslash in\; N(C)$, then any subset of $J$ is also in $N(C)$, making $N(C)$ an abstract simplicial complex. Hence N(C) is often called the nerve complex of $C$.

1. Let X be the circle $S^1$ and $C\; =\; \backslash \{U\_1,\; U\_2\backslash \}$, where $U\_1$ is an arc covering the upper half of $S^1$ and $U\_2$ is an arc covering its lower half, with some overlap at both sides (they must overlap at both sides in order to cover all of $S^1$). Then $N(C)\; =\; \backslash \{\; \backslash \{1\backslash \},\; \backslash \{2\backslash \},\; \backslash \{1,2\backslash \}\; \backslash \}$, which is an abstract 1-simplex.
2. Let X be the circle $S^1$ and $C\; =\; \backslash \{U\_1,\; U\_2,\; U\_3\backslash \}$, where each $U\_i$ is an arc covering one third of $S^1$, with some overlap with the adjacent $U\_i$. Then $N(C)\; =\; \backslash \{\; \backslash \{1\backslash \},\; \backslash \{2\backslash \},\; \backslash \{3\backslash \},\; \backslash \{1,2\backslash \},\; \backslash \{2,3\backslash \},\; \backslash \{3,1\backslash \}\; \backslash \}$. Note that {1,2,3} is not in $N(C)$ since the common intersection of all three sets is empty; so $N(C)$ is an unfilled triangle.

Given an open cover $C=\backslash \{U\_i:\; i\backslash in\; I\backslash \}$ of a topological space $X$, or more generally a cover in a site, we can consider the pairwise fibre products $U\_\{ij\}=U\_i\backslash times\_XU\_j$, which in the case of a topological space are precisely the intersections $U\_i\backslash cap\; U\_j$. The collection of all such intersections can be referred to as $C\backslash times\_X\; C$ and the triple intersections as $C\backslash times\_X\; C\backslash times\_X\; C$.

By considering the natural maps $U\_\{ij\}\backslash to\; U\_i$ and $U\_i\backslash to\; U\_\{ii\}$, we can construct a simplicial object $S(C)\_\backslash bullet$ defined by $S(C)\_n=C\backslash times\_X\backslash cdots\backslash times\_XC$, n-fold fibre product. This is the Čech nerve.{{Cite web|title=Čech nerve in nLab|url=https://ncatlab.org/nlab/show/%C4%8Cech+nerve|access-date=2020-08-07|website=ncatlab.org}}

By taking connected components we get a simplicial set, which we can realise topologically: $|S(\backslash pi\_0(C))|$.

The nerve complex $N(C)$ is a simple combinatorial object. Often, it is much simpler than the underlying topological space (the union of the sets in $C$). Therefore, a natural question is whether the topology of $N(C)$ is equivalent to the topology of $\backslash bigcup\; C$.

In general, this need not be the case. For example, one can cover any n-sphere with two contractible sets $U\_1$ and $U\_2$ that have a non-empty intersection, as in example 1 above. In this case, $N(C)$ is an abstract 1-simplex, which is similar to a line but not to a sphere.

However, in some cases $N(C)$ does reflect the topology of X. For example, if a circle is covered by three open arcs, intersecting in pairs as in Example 2 above, then $N(C)$ is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle.{{Cite book|last1=Artin|first1=Michael|author1-link=Michael Artin|last2=Mazur|first2=Barry|author2-link=Barry Mazur|date=1969|title=Etale Homotopy|series=Lecture Notes in Mathematics|volume=100| doi=10.1007/bfb0080957|isbn=978-3-540-04619-6|issn=0075-8434}}

A nerve theorem (or nerve lemma) is a theorem that gives sufficient conditions on C guaranteeing that $N(C)$ reflects, in some sense, the topology of $\backslash bigcup\; C$. A functorial nerve theorem is a nerve theorem that is functorial in an appropriate sense, which is, for example, crucial in topological data analysis.{{Cite journal|last1=Bauer|first1=Ulrich|last2=Kerber|first2=Michael|last3=Roll|first3=Fabian|last4=Rolle|first4=Alexander|date=2023|title=A unified view on the functorial nerve theorem and its variations|journal=Expositiones Mathematicae|volume=41 |issue=4 | language=en|doi=10.1016/j.exmath.2023.04.005|arxiv=2203.03571}}

The basic nerve theorem of Jean Leray says that, if any intersection of sets in $N(C)$ is contractible (equivalently: for each finite $J\backslash subset\; I$ the set $\backslash bigcap\_\{i\backslash in\; J\}\; U\_i$ is either empty or contractible; equivalently: C is a good open cover), then $N(C)$ is homotopy-equivalent to $\backslash bigcup\; C$.

There is a discrete version, which is attributed to Borsuk.{{Cite journal |last=Borsuk |first=Karol |date=1948 |title=On the imbedding of systems of compacta in simplicial complexes |url=https://eudml.org/doc/213158 |journal=Fundamenta Mathematicae |volume=35 |issue=1 |pages=217–234 |doi=10.4064/fm-35-1-217-234 |issn=0016-2736|doi-access=free }}{{Rp|page=81|location=Thm.4.4.4}} Let K₁,...,K_n be abstract simplicial complexes, and denote their union by K. Let U_i = ||K_i|| = the geometric realization of K_i, and denote the nerve of {U₁, ... , U_n } by N.

If, for each nonempty $J\backslash subset\; I$, the intersection $\backslash bigcap\_\{i\backslash in\; J\}\; U\_i$ is either empty or contractible, then N is homotopy-equivalent to K.

A stronger theorem was proved by Anders Bjorner.{{Cite journal |last=Björner |first=Anders |authorlink=Anders Björner|date=2003-04-01 |title=Nerves, fibers and homotopy groups |journal=Journal of Combinatorial Theory|series=Series A |language=en |volume=102 |issue=1 |pages=88–93 |doi=10.1016/S0097-3165(03)00015-3 |doi-access=free |issn=0097-3165}} if, for each nonempty $J\backslash subset\; I$, the intersection $\backslash bigcap\_\{i\backslash in\; J\}\; U\_i$ is either empty or N-connected space, then for every j ≤ k, the j-th homotopy group of N is isomorphic to the j-th homotopy group of K. In particular, N is k-connected if-and-only-if K is k-connected.

Another nerve theorem relates to the Čech nerve above: if $X$ is compact and all intersections of sets in C are contractible or empty, then the space $|S(\backslash pi\_0(C))|$ is homotopy-equivalent to $X$.{{nlab|id=nerve+theorem|title=Nerve theorem}}

The following nerve theorem uses the homology groups of intersections of sets in the cover.{{Cite journal|last=Meshulam|first=Roy|date=2001-01-01|title=The Clique Complex and Hypergraph Matching|journal=Combinatorica| language=en|volume=21|issue=1|pages=89–94|doi=10.1007/s004930170006|s2cid=207006642|issn=1439-6912}} For each finite $J\backslash subset\; I$, denote $H\_\{J,j\}\; :=\; \backslash tilde\{H\}\_j(\backslash bigcap\_\{i\backslash in\; J\}\; U\_i)=$ the j-th reduced homology group of $\backslash bigcap\_\{i\backslash in\; J\}\; U\_i$.

If H_J,j is the trivial group for all J in the k-skeleton of N(C) and for all j in {0, ..., k-dim(J)}, then N(C) is "homology-equivalent" to X in the following sense:

• $\backslash tilde\{H\}\_j(N(C))\; \backslash cong\; \backslash tilde\{H\}\_j(X)$ for all j in {0, ..., k};
• if $\backslash tilde\{H\}\_\{k+1\}(N(C))\backslash not\backslash cong\; 0$ then $\backslash tilde\{H\}\_\{k+1\}(X)\backslash not\backslash cong\; 0$ .

{{DEFAULTSORT:Nerve Of A Covering}}

Category:Topology

Category:Simplicial sets

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