Nerve complex

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 ^[1] 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.^[2]

Let ${\displaystyle I}${\displaystyle I} be a set of indices and ${\displaystyle C}${\displaystyle C} be a family of sets ${\displaystyle (U_{i})_{i\in I}}${\displaystyle (U_{i})_{i\in I}}. The nerve of ${\displaystyle C}${\displaystyle C} is a set of finite subsets of the index set ${\displaystyle I}${\displaystyle I}. It contains all finite subsets ${\displaystyle J\subseteq I}${\displaystyle J\subseteq I} such that the intersection of the ${\displaystyle U_{i}}${\displaystyle U_{i}} whose subindices are in ${\displaystyle J}${\displaystyle J} is non-empty:^{[3] : 81}

${\displaystyle N(C):={\bigg \{}J\subseteq I:\bigcap _{j\in J}U_{j}\neq \varnothing ,J{\text{ finite set}}{\bigg \}}.}${\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 ${\displaystyle (U_{i})_{i\in I}}${\displaystyle (U_{i})_{i\in I}} are open subsets of some topological space ${\displaystyle X}${\displaystyle X}.

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

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

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

By considering the natural maps ${\displaystyle U_{ij}\to U_{i}}${\displaystyle U_{ij}\to U_{i}} and ${\displaystyle U_{i}\to U_{ii}}${\displaystyle U_{i}\to U_{ii}}, we can construct a simplicial object ${\displaystyle S(C)_{\bullet }}${\displaystyle S(C)_{\bullet }} defined by ${\displaystyle S(C)_{n}=C\times _{X}\cdots \times _{X}C}${\displaystyle S(C)_{n}=C\times _{X}\cdots \times _{X}C}, n-fold fibre product. This is the Čech nerve.^[4]

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

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

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

However, in some cases ${\displaystyle N(C)}${\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 ${\displaystyle N(C)}${\displaystyle N(C)} is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle.^[5]

A nerve theorem (or nerve lemma) is a theorem that gives sufficient conditions on C guaranteeing that ${\displaystyle N(C)}${\displaystyle N(C)} reflects, in some sense, the topology of ${\displaystyle \bigcup C}${\displaystyle \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.^[6]

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

There is a discrete version, which is attributed to Borsuk.^{[7] [3] : 81, 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 ${\displaystyle J\subset I}${\displaystyle J\subset I}, the intersection ${\displaystyle \bigcap _{i\in J}U_{i}}${\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.^[8] if, for each nonempty ${\displaystyle J\subset I}${\displaystyle J\subset I}, the intersection ${\displaystyle \bigcap _{i\in J}U_{i}}${\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.

Another nerve theorem relates to the Čech nerve above: if ${\displaystyle X}${\displaystyle X} is compact and all intersections of sets in C are contractible or empty, then the space ${\displaystyle |S(\pi _{0}(C))|}${\displaystyle |S(\pi _{0}(C))|} is homotopy-equivalent to ${\displaystyle X}${\displaystyle X}.^[9]

The following nerve theorem uses the homology groups of intersections of sets in the cover.^[10] For each finite ${\displaystyle J\subset I}${\displaystyle J\subset I}, denote ${\displaystyle H_{J,j}:={\tilde {H}}_{j}(\bigcap _{i\in J}U_{i})=}${\displaystyle H_{J,j}:={\tilde {H}}_{j}(\bigcap _{i\in J}U_{i})=} the j-th reduced homology group of ${\displaystyle \bigcap _{i\in J}U_{i}}${\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:

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

• Hypercovering

• Topology
• Simplicial sets
• Families of sets