Nerve of a covering

Constructing the nerve of an open good cover containing 3 sets in the plane.

In topology, the nerve of an open covering is a construction of an abstract simplicial complex from an open covering of a topological space X that captures many of the interesting topological properties in an algorithmic or combinatorial way. 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.^[2]

Alexandrov's definition[]

Let ${\displaystyle X}$ be a topological space, ${\displaystyle I}$ be an index set and ${\displaystyle C}$ be a family of open subsets ${\displaystyle U_{i}}$ of ${\displaystyle X}$ indexed by ${\displaystyle i\in I}$. 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:

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

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, 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.

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.^[3]

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

Nerve theorems[]

In general, the complex ${\displaystyle N(C)}$ need not reflect the topology of X accurately. For example, we 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, 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.^[4]

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 X.

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 X.^[5]

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 ${\displaystyle X}$.^[6]

Homological nerve theorem[]

The following nerve theorem uses the homology groups of intersections of sets in the cover.^[7] 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[]