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.
is non-empty:[3]: 81 In Alexandrov's original definition, the sets
are open subsets of some topological space
an abstract simplicial complex.
, or more generally a cover in a site, we can consider the pairwise fibre products
, which in the case of a topological space are precisely the intersections
, we can construct a simplicial object
, n-fold fibre product.
[4] By taking connected components we get a simplicial set, which we can realise topologically:
Often, it is much simpler than the underlying topological space (the union of the sets in
For example, one can cover any n-sphere with two contractible sets
is an abstract 1-simplex, which is similar to a line but not to a sphere.
For example, if a circle is covered by three open arcs, intersecting in pairs as in Example 2 above, then
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
reflects, in some sense, the topology of
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
is contractible (equivalently: for each finite
is either empty or contractible; equivalently: C is a good open cover), then
There is a discrete version, which is attributed to Borsuk.
[7][3]: 81, Thm.4.4.4 Let K1,...,Kn be abstract simplicial complexes, and denote their union by K. Let Ui = ||Ki|| = the geometric realization of Ki, and denote the nerve of {U1, ... , Un } by N. If, for each nonempty
is either empty or contractible, then N is homotopy-equivalent to K. A stronger theorem was proved by Anders Bjorner.
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.
is compact and all intersections of sets in C are contractible or empty, then the space
[9] The following nerve theorem uses the homology groups of intersections of sets in the cover.
the j-th reduced homology group of
If HJ,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:
