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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 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.

Basic definition

Let ${\displaystyle I}$ be a set of indices and ${\displaystyle C}$ be a family of sets ${\displaystyle (U_{i})_{i\in I}}$. The nerve of ${\displaystyle C}$ is a set of finite subsets of the index set ${\displaystyle I}$. It contains all finite subsets ${\displaystyle J\subseteq I}$ such that the intersection of the ${\displaystyle U_{i}}$ whose subindices are in ${\displaystyle J}$ is non-empty:

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

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

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

Examples

1. Let X be the circle ${\displaystyle S^{1}}$ and ${\displaystyle C=\{U_{1},U_{2}\}}$, where ${\displaystyle U_{1}}$ is an arc covering the upper half of ${\displaystyle S^{1}}$ and ${\displaystyle U_{2}}$ 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 ${\displaystyle S^{1}}$). Then ${\displaystyle N(C)=\{\{1\},\{2\},\{1,2\}\}}$, which is an abstract 1-simplex.
2. Let X be the circle ${\displaystyle S^{1}}$ and ${\displaystyle C=\{U_{1},U_{2},U_{3}\}}$, where each ${\displaystyle U_{i}}$ is an arc covering one third of ${\displaystyle S^{1}}$, with some overlap with the adjacent ${\displaystyle U_{i}}$. Then ${\displaystyle N(C)=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{3,1\}\}}$. Note that {1,2,3} is not in ${\displaystyle N(C)}$ since the common intersection of all three sets is empty; so ${\displaystyle N(C)}$ is an unfilled triangle.

The Čech nerve

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

By considering the natural maps ${\displaystyle U_{ij}\to U_{i}}$ and ${\displaystyle U_{i}\to U_{ii}}$, we can construct a simplicial object ${\displaystyle S(C)_{\bullet }}$ defined by ${\displaystyle S(C)_{n}=C\times _{X}\cdots \times _{X}C}$, n-fold fibre product. This is the Čech nerve.

By taking connected components we get a simplicial set, which we can realise topologically: ${\displaystyle |S(\pi _{0}(C))|}$.

Nerve theorems

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

In general, this need not be the case. For example, one can cover any n-sphere with two contractible sets ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ that have a non-empty intersection, as in example 1 above. In this case, ${\displaystyle N(C)}$ is an abstract 1-simplex, which is similar to a line but not to a sphere.

However, in some cases ${\displaystyle N(C)}$ 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 ${\displaystyle N(C)}$ is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle.

A nerve theorem (or nerve lemma) is a theorem that gives sufficient conditions on C guaranteeing that ${\displaystyle N(C)}$ reflects, in some sense, the topology of ${\displaystyle \bigcup C}$. A functorial nerve theorem is a nerve theorem that is functorial in an approriate sense, which is, for example, crucial in topological data analysis.

Leray's nerve theorem

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

Borsuk's nerve theorem

There is a discrete version, which is attributed to Borsuk. 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 ${\displaystyle J\subset I}$, the intersection ${\displaystyle \bigcap _{i\in J}U_{i}}$ is either empty or contractible, then N is homotopy-equivalent to K.

A stronger theorem was proved by Anders Bjorner. if, for each nonempty ${\displaystyle J\subset I}$, the intersection ${\displaystyle \bigcap _{i\in J}U_{i}}$ 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 ${\displaystyle X}$ is compact and all intersections of sets in C are contractible or empty, then the space ${\displaystyle |S(\pi _{0}(C))|}$ is homotopy-equivalent to ${\displaystyle X}$.

Homological nerve theorem

The following nerve theorem uses the homology groups of intersections of sets in the cover. For each finite ${\displaystyle J\subset I}$, denote ${\displaystyle H_{J,j}:={\tilde {H}}_{j}(\bigcap _{i\in J}U_{i})=}$ the j-th reduced homology group of ${\displaystyle \bigcap _{i\in J}U_{i}}$.

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:

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

See also

• Hypercovering

References

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2. Eilenberg, Samuel; Steenrod, Norman (1952-12-31). Foundations of Algebraic Topology. Princeton: Princeton University Press. doi:10.1515/9781400877492. ISBN 978-1-4008-7749-2.
3. ^ Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
4. "Čech nerve in nLab". ncatlab.org. Retrieved 2020-08-07.
5. Artin, Michael; Mazur, Barry (1969). Etale Homotopy. Lecture Notes in Mathematics. Vol. 100. doi:10.1007/bfb0080957. ISBN 978-3-540-04619-6. ISSN 0075-8434.
6. Bauer, Ulrich; Kerber, Michael; Roll, Fabian; Rolle, Alexander (2023). "A unified view on the functorial nerve theorem and its variations". Expositiones Mathematicae. arXiv:2203.03571. doi:10.1016/j.exmath.2023.04.005.
7. Borsuk, Karol (1948). "On the imbedding of systems of compacta in simplicial complexes". Fundamenta Mathematicae. 35 (1): 217–234. doi:10.4064/fm-35-1-217-234. ISSN 0016-2736.
8. Björner, Anders (2003-04-01). "Nerves, fibers and homotopy groups". Journal of Combinatorial Theory. Series A. 102 (1): 88–93. doi:10.1016/S0097-3165(03)00015-3. ISSN 0097-3165.
9. Nerve theorem at the nLab
10. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642.

• Topology
• Simplicial sets
• Families of sets