Nerve complex explained

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]

Basic definition

Let

I be a set of indices and C be a family of sets (U _{i)} _{i\in} . The nerve of C is a set of finite subsets of the index set I . It contains all finite subsets J\subseteq I such that the intersection of the U _{i} whose subindices are in J is non-empty:^[3] N(C) : = \ {J\subseteq I: c ap _{j\in} U _{j} ≠ \varnothing, J finiteset \ }. In Alexandrov's original definition, the sets (U _{i)} _{i\in} are open subsets of some topological space X .

The set

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

Examples

1. Let X be the circle

S ^{1} and C = \{U _{1,} U _{2\}} , where U _{1} is an arc covering the upper half of S ^{1} and 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 S ^{1} ). Then N(C) = \{ \{ 1 \}, \{ 2 \}, \{ 1 , 2 \} \} , which is an abstract 1-simplex.

1. Let X be the circle

S ^{1} and C = \{U _{1,} U _{2,} U _{3\}} , where each U _{i} is an arc covering one third of S ^{1} , with some overlap with the adjacent U _{i} . Then N(C) = \{ \{ 1 \}, \{ 2 \}, \{ 3 \}, \{ 1 , 2 \}, \{ 2 , 3 \}, \{ 3 , 1 \} \} . Note that is not in N(C) since the common intersection of all three sets is empty; so N(C) is an unfilled triangle.

The Čech nerve

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

By considering the natural maps

U _{ij} \to U _{i} and U _{i\to} U _{ii} , we can construct a simplicial object S(C) _{\bullet} defined by S(C) _{n=C x} _{X … x} _{XC} , n-fold fibre product. This is the Čech nerve.^[4]

By taking connected components we get a simplicial set, which we can realise topologically:

|S(\pi _{0(C))|} .

Nerve theorems

The nerve complex

N(C) is a simple combinatorial object. Often, it is much simpler than the underlying topological space (the union of the sets in C ). Therefore, a natural question is whether the topology of N(C) is equivalent to the topology of c up C .

In general, this need not be the case. For example, one can cover any n-sphere with two contractible sets

U _{1} and 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

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.^[5]

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 c up 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]

Leray's nerve theorem

The basic nerve theorem of Jean Leray says that, if any intersection of sets in

N(C) is contractible (equivalently: for each finite J\subset I the set c ap _{i\in} U _{i} is either empty or contractible; equivalently: C is a good open cover), then N(C) is homotopy-equivalent to c up C .

Borsuk's nerve theorem

There is a discrete version, which is attributed to Borsuk.^[7] 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 by N.

If, for each nonempty

J\subset I , the intersection c ap _{i\in} 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

J\subset I , the intersection c ap _{i\in} 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.

Čech nerve theorem

Another nerve theorem relates to the Čech nerve above: if

X is compact and all intersections of sets in C are contractible or empty, then the space |S(\pi _{0(C))|} is homotopy-equivalent to X .

Homological nerve theorem

The following nerve theorem uses the homology groups of intersections of sets in the cover.^[9] For each finite

J\subset I , denote H _{J,j} : = \tilde{H} _{j(cap} _{i\in} U _{i)=} the j-th reduced homology group of c ap _{i\in} 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, then N(C) is "homology-equivalent" to X in the following sense:

\tilde{H} _{j(N(C))} \cong \tilde{H} _{j(X)} for all j in ;

• if

\tilde{H} _{k+1} (N(C))\not\cong 0 then \tilde{H} _{k+1} (X)\not\cong 0 .

See also

• Hypercovering

References

1. Aleksandroff . P. S. . Pavel Alexandrov . 1928 . Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung . . 98 . 617–635 . 10.1007/BF01451612 . 119590045.
2. Book: Eilenberg . Samuel . Foundations of Algebraic Topology . Steenrod . Norman . 1952-12-31 . . 978-1-4008-7749-2 . Princeton . 10.1515/9781400877492 . Samuel Eilenberg . Norman Steenrod.
3. , Section 4.3
4. Web site: Čech nerve in nLab. 2020-08-07. ncatlab.org.
5. Book: Artin. Michael. Michael Artin. Mazur. Barry. Barry Mazur. 1969. Etale Homotopy. Lecture Notes in Mathematics. 100. 10.1007/bfb0080957. 978-3-540-04619-6. 0075-8434.
6. Bauer. Ulrich. Kerber. Michael. Roll. Fabian. Rolle. Alexander. 2023. A unified view on the functorial nerve theorem and its variations. Expositiones Mathematicae. en. 10.1016/j.exmath.2023.04.005. 2203.03571.
7. Borsuk . Karol . 1948 . On the imbedding of systems of compacta in simplicial complexes . Fundamenta Mathematicae . 35 . 1 . 217–234 . 10.4064/fm-35-1-217-234 . 0016-2736. free .
8. Björner . Anders . Anders Björner. 2003-04-01 . Nerves, fibers and homotopy groups . Journal of Combinatorial Theory. Series A . en . 102 . 1 . 88–93 . 10.1016/S0097-3165(03)00015-3 . free . 0097-3165.
9. Meshulam. Roy. 2001-01-01. The Clique Complex and Hypergraph Matching. Combinatorica. en. 21. 1. 89–94. 10.1007/s004930170006. 207006642. 1439-6912.