Nerve of a covering

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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[edit]

Let be a topological space, be an index set and be a family of open subsets of indexed by . The nerve of is a set of finite subsets of the index-set . It contains all finite subsets such that the intersection of the whose subindices are in is non-empty:

may contain singletons (elements such that is non-empty), pairs (pairs of elements such that ), triplets, and so on. If , then any subset of is also in , making an abstract simplicial complex, often called the nerve complex of .

Examples[edit]

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 Ui is an arc covering one third of S1, with some overlap with the adjacent Ui. 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[edit]

Given an open cover of a topological space , 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 . The collection of all such intersections can be referred to as and the triple intersections as .

By considering the natural maps and , we can construct a simplicial object defined by , n-fold fibre product. This is the Čech nerve. [3]

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

Nerve theorems[edit]

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.

[4]

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 the set is either empty or contractible; equivalently: C is a good open cover), then N(C) is homotopy-equivalent to X.[5]

Another nerve theorem relates to the Čech nerve above: if is compact and all intersections of sets in C are contractible or empty, then the space is homotopy-equivalent to .[6]

Homological nerve theorem[edit]

The following nerve theorem uses the homology groups of intersections of sets in the cover.[7] For each finite , denote 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:

  • for all j in {0, ..., k};
  • if then .

See also[edit]

References[edit]

  1. ^ Aleksandroff, P. S. (1928). "Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung". Mathematische Annalen. 98: 617–635. doi:10.1007/BF01451612. S2CID 119590045.
  2. ^ Eilenberg, Samuel; Steenrod, Norman (1952-12-31). Foundations of Algebraic Topology. Princeton: Princeton University Press. doi:10.1515/9781400877492. ISBN 978-1-4008-7749-2.
  3. ^ "Čech nerve in nLab". ncatlab.org. Retrieved 2020-08-07.
  4. ^ Artin, M.; Mazur, B. (1969). "Etale Homotopy". Lecture Notes in Mathematics. 100. doi:10.1007/bfb0080957. ISBN 978-3-540-04619-6. ISSN 0075-8434.
  5. ^ 1969-, Ghrist, Robert W. (2014). Elementary applied topology (Edition 1.0 ed.). [United States]. ISBN 9781502880857. OCLC 899283974.CS1 maint: numeric names: authors list (link)
  6. ^ Nerve theorem in nLab
  7. ^ Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642.