Nerve complex

{{Short description|Complex recording the pattern of intersections between a topological family's sets}} [[File:Constructing nerve.png|thumb|Constructing the nerve of an open good cover containing 3 sets in the plane.]]

In topology, the '''nerve complex''' of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov<ref>{{cite journal |last=Aleksandroff |first=P. S. |author-link=Pavel Alexandrov |year=1928 |title=Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung |journal=Mathematische Annalen |volume=98 |pages=617–635 |doi=10.1007/BF01451612 |s2cid=119590045}}</ref> and now has many variants and generalisations, among them the '''Čech nerve''' of a cover, which in turn is generalised by hypercoverings. It captures many of the interesting topological properties in an algorithmic or combinatorial way.<ref>{{Cite book |last1=Eilenberg |first1=Samuel |title=Foundations of Algebraic Topology |last2=Steenrod |first2=Norman |date=1952-12-31 |publisher=Princeton University Press |isbn=978-1-4008-7749-2 |location=Princeton |doi=10.1515/9781400877492 |author1-link=Samuel Eilenberg |author2-link=Norman Steenrod}}</ref>

==Basic definition== Let <math>I</math> be a set of indices and <math>C</math> be a family of sets <math>(U_i)_{i\in I}</math>. The '''nerve''' of <math>C</math> is a set of finite subsets of the index set ''<math>I</math>''. It contains all finite subsets <math>J\subseteq I</math> such that the intersection of the <math>U_i</math> whose subindices are in <math>J</math> is non-empty:''<ref name=":0">{{Cite Matousek 2007}}, Section 4.3</ref>{{Rp|page=81|location=}}'' :<math>N(C) := \bigg\{J\subseteq I: \bigcap_{j\in J}U_j \neq \varnothing, J \text{ finite set} \bigg\}.</math> In Alexandrov's original definition, the sets <math>(U_i)_{i\in I}</math> are open subsets of some topological space <math>X</math>.

The set <math>N(C)</math> may contain singletons (elements <math>i \in I</math> such that <math>U_i</math> is non-empty), pairs (pairs of elements <math>i,j \in I</math> such that <math>U_i \cap U_j \neq \emptyset</math>), triplets, and so on. If <math>J \in N(C)</math>, then any subset of <math>J</math> is also in <math>N(C)</math>, making <math>N(C)</math> an abstract simplicial complex. Hence N(C) is often called the '''nerve complex''' of <math>C</math>.

==Examples== # Let ''X'' be the circle <math>S^1</math> and <math>C = \{U_1, U_2\}</math>, where <math>U_1</math> is an arc covering the upper half of <math>S^1</math> and <math>U_2</math> 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 <math>S^1</math>). Then <math>N(C) = \{ \{1\}, \{2\}, \{1,2\} \}</math>, which is an abstract 1-simplex. # Let ''X'' be the circle <math>S^1</math> and <math>C = \{U_1, U_2, U_3\}</math>, where each <math>U_i</math> is an arc covering one third of <math>S^1</math>, with some overlap with the adjacent <math>U_i</math>. Then <math>N(C) = \{ \{1\}, \{2\}, \{3\}, \{1,2\}, \{2,3\}, \{3,1\} \}</math>. Note that {1,2,3} is not in <math>N(C)</math> since the common intersection of all three sets is empty; so <math>N(C)</math> is an unfilled triangle. # Let <math>P</math> be a finite set of points in general position (no four points are cocircular) in <math>\mathbb{R}^2</math>, and let <math>C = \{V_p\}_{p \in P}</math> be the set of cells of the Voronoi diagram of <math>P</math>. Then the nerve <math>N(C)</math> is the Delaunay triangulation of <math>P</math>.<ref>{{cite book |last=Virk |first=Žiga |title=Introduction to Persistent Homology |year=2022 |publisher=Faculty of Computer and Information Science, University of Ljubljana |url=https://zigavirk.gitlab.io/PhBook.pdf |pages=26–27,64 |access-date=2026-02-14}}</ref> Note that the Voronoi cells are closed sets, so this is a nerve of a closed cover rather than an open one; however, the definition of the nerve applies to arbitrary set families.

==The Čech nerve== Given an open cover <math>C=\{U_i: i\in I\}</math> of a topological space <math>X</math>, or more generally a cover in a site, we can consider the pairwise fibre products <math>U_{ij}=U_i\times_XU_j</math>, which in the case of a topological space are precisely the intersections <math>U_i\cap U_j</math>. The collection of all such intersections can be referred to as <math>C\times_X C</math> and the triple intersections as <math>C\times_X C\times_X C</math>.

By considering the natural maps <math>U_{ij}\to U_i</math> and <math>U_i\to U_{ii}</math>, we can construct a simplicial object <math>S(C)_\bullet</math> defined by <math>S(C)_n=C\times_X\cdots\times_XC</math>, n-fold fibre product. This is the '''Čech nerve.'''<ref>{{Cite web|title=Čech nerve in nLab|url=https://ncatlab.org/nlab/show/%C4%8Cech+nerve|access-date=2020-08-07|website=ncatlab.org}}</ref>

By taking connected components we get a simplicial set, which we can realise topologically: <math>|S(\pi_0(C))|</math>.

==Nerve theorems== The nerve complex <math>N(C)</math> is a simple combinatorial object. Often, it is much simpler than the underlying topological space (the union of the sets in <math>C</math>). Therefore, a natural question is whether the topology of <math>N(C)</math> is equivalent to the topology of <math>\bigcup C</math>.

In general, this need not be the case. For example, one can cover any ''n''-sphere with two contractible sets <math>U_1</math> and <math>U_2</math> that have a non-empty intersection, as in example 1 above. In this case, <math>N(C)</math> is an abstract 1-simplex, which is similar to a line but not to a sphere.

However, in some cases <math>N(C)</math> 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 <math>N(C)</math> is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle.<ref>{{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}}</ref>

A '''nerve theorem''' (or '''nerve lemma''') is a theorem that gives sufficient conditions on ''C'' guaranteeing that <math>N(C)</math> reflects, in some sense, the topology of ''<math>\bigcup C</math>''. A '''functorial nerve theorem''' is a nerve theorem that is functorial in an appropriate sense, which is, for example, crucial in topological data analysis.<ref>{{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}}</ref>

=== Leray's nerve theorem === The basic nerve theorem of Jean Leray says that, if any intersection of sets in <math>N(C)</math> is contractible (equivalently: for each finite <math>J\subset I</math> the set <math>\bigcap_{i\in J} U_i</math> is either empty or contractible; equivalently: ''C'' is a good open cover), then <math>N(C)</math> is homotopy-equivalent to ''<math>\bigcup C</math>''.

=== Borsuk's nerve theorem === There is a discrete version, which is attributed to Borsuk.<ref>{{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 }}</ref>''<ref name=":0" />{{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 <math>J\subset I</math>, the intersection <math>\bigcap_{i\in J} U_i</math> is either empty or contractible, then ''N'' is homotopy-equivalent to ''K''.

A stronger theorem was proved by Anders Bjorner.<ref>{{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}}</ref> If, for each nonempty <math>J\subset I</math>, the intersection <math>\bigcap_{i\in J} U_i</math> is either empty or (k-|J|+1)-connected, 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.

=== Čech nerve theorem === Another nerve theorem relates to the Čech nerve above: if <math>X</math> is compact and all intersections of sets in ''C'' are contractible or empty, then the space <math>|S(\pi_0(C))|</math> is homotopy-equivalent to <math>X</math>.<ref>{{nlab|id=nerve+theorem|title=Nerve theorem}}</ref>

=== Homological nerve theorem === The following nerve theorem uses the homology groups of intersections of sets in the cover.<ref name=":3">{{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}}</ref> For each finite <math>J\subset I</math>, denote <math>H_{J,j} := \tilde{H}_j(\bigcap_{i\in J} U_i)=</math> the ''j''-th reduced homology group of <math>\bigcap_{i\in J} U_i</math>.

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:

* <math>\tilde{H}_j(N(C)) \cong \tilde{H}_j(X)</math> for all ''j'' in {0, ..., ''k''}; * if <math>\tilde{H}_{k+1}(N(C))\not\cong 0</math> then <math>\tilde{H}_{k+1}(X)\not\cong 0</math> .

== References == <references/>

{{DEFAULTSORT:Nerve Of A Covering}} Category:Topology Category:Simplicial sets Category:Families of sets