Nerve complex

File:Constructing nerve.png

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>Template:Cite journal</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>Template:Cite book</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">Template:Cite Matousek 2007, Section 4.3</ref>Template:Rp

<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

1. 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.
2. 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.

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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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>Template:Cite book</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>Template:Cite journal</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>Template:Cite journal</ref><ref name=":0" />Template:Rp 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>Template:Cite journal</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>Template:Nlab</ref>

Homological nerve theorem

The following nerve theorem uses the homology groups of intersections of sets in the cover.<ref name=":3">Template:Cite journal</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

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