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 ${\displaystyle I}$ 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:^[3]: 81

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

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 ${\displaystyle C}$.

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.

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.^[4]

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

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.^[5]

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.^[6]

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 ${\displaystyle \bigcup C}$.

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 ${\displaystyle J\subset I}$, 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.^[8] 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.

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 ${\displaystyle X}$.^[9]

The following nerve theorem uses the homology groups of intersections of sets in the cover.^[10] 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 ${\displaystyle {\tilde {H}}_{k+1}(X)\not \cong 0}$ .

• Hypercovering