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 and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings.

Alexandrov's definition

Let [math]\displaystyle{ X }[/math] be a topological space, [math]\displaystyle{ I }[/math] be an index set and [math]\displaystyle{ C }[/math] be a family of open subsets [math]\displaystyle{ U_i }[/math] of [math]\displaystyle{ X }[/math] indexed by [math]\displaystyle{ i \in I }[/math]. The nerve of [math]\displaystyle{ C }[/math] is a set of finite subsets of the index-set [math]\displaystyle{ I }[/math]. It contains all finite subsets [math]\displaystyle{ J\subseteq I }[/math] such that the intersection of the [math]\displaystyle{ U_i }[/math] whose subindices are in [math]\displaystyle{ J }[/math] is non-empty:

[math]\displaystyle{ N(C)  := \bigg\{J\subseteq I: \bigcap_{j\in J}U_j \neq \varnothing, J \text{ finite set} \bigg\}. }[/math]

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

Examples

1. Let X be the circle S¹ and C = {U₁, U₂}, where U₁ is an arc covering the upper half of S¹ and U₂ 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 S¹). Then N(C) = { {1}, {2}, {1,2} }, which is an abstract 1-simplex.

2. Let X be the circle S¹ and C = {U₁, U₂, U₃}, where each U_i is an arc covering one third of S¹, with some overlap with the adjacent U_i. Then N(C) = { {1}, {2}, {3}, {1,2}, {2,3}, {3,1} }. Note that {1,2,3} is not in N(C) since the common intersection of all three sets is empty.

The Čech nerve

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

By considering the natural maps [math]\displaystyle{ U_{ij}\to U_i }[/math] and [math]\displaystyle{ U_i\to U_{ii} }[/math], we can construct a simplicial object [math]\displaystyle{ S(C)_\bullet }[/math] defined by [math]\displaystyle{ S(C)_n=C\times_X\cdots\times_XC }[/math], n-fold fibre product. This is the Čech nerve.

By taking connected components we get a simplicial set, which we can realise topologically: [math]\displaystyle{ |S(\pi_0(C))| }[/math].

Nerve theorems

In general, the complex N(C) need not reflect the topology of X accurately. For example, we can cover any n-sphere with two contractible sets U₁ and U₂ 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 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 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 N(C) reflects, in some sense, the topology of X.

The basic nerve theorem of Leray says that, if any intersection of sets in N(C) is contractible (equivalently: for each finite [math]\displaystyle{ J\subset I }[/math] the set [math]\displaystyle{ \bigcap_{i\in J} U_i }[/math] is either empty or contractible; equivalently: C is a good open cover), then N(C) is homotopy-equivalent to X.

Another nerve theorem relates to the Čech nerve above: if [math]\displaystyle{ X }[/math] is compact and all intersections of sets in C are contractible or empty, then the space [math]\displaystyle{ |S(\pi_0(C))| }[/math] is homotopy-equivalent to [math]\displaystyle{ X }[/math].

Homological nerve theorem

The following nerve theorem uses the homology groups of intersections of sets in the cover. For each finite [math]\displaystyle{ J\subset I }[/math], denote [math]\displaystyle{ H_{J,j} := \tilde{H}_j(\bigcap_{i\in J} U_i)= }[/math] the j-th reduced homology group of [math]\displaystyle{ \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]\displaystyle{ \tilde{H}_j(N(C)) \cong \tilde{H}_j(X) }[/math] for all j in {0, ..., k};
• if [math]\displaystyle{ \tilde{H}_{k+1}(N(C))\not\cong 0 }[/math] then [math]\displaystyle{ \tilde{H}_{k+1}(X)\not\cong 0 }[/math] .

See also

• Hypercovering