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Nerve complex
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==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|''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 space|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 cover|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 [[Karol Borsuk|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 ''K<sub>1</sub>,...,K<sub>n</sub>'' be [[Abstract simplicial complex|abstract simplicial complexes]], and denote their union by ''K''. Let ''U<sub>i</sub>'' = ||''K<sub>i</sub>||'' = the [[Abstract simplicial complex|geometric realization]] of ''K<sub>i</sub>'', and denote the nerve of {''U<sub>1</sub>'', ... , ''U<sub>n</sub>'' } 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 Björner|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 [[N-connected space|(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<sub>J,j</sub>'' 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> .
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