Nerve of a covering

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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.

Alexandrov's definiton[edit]

Given an index set I, and open sets Ui contained in X, the nerve N is the set of finite subsets of I defined as follows:

  • a finite set JI belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty. That is, if and only if

Obviously, if J belongs to N, then any of its subsets is also in N. Therefore N is an abstract simplicial complex.

In general, the complex N need not reflect the topology of X accurately. For example we can cover any n-sphere with two contractible sets U and V, in such a way that N is an abstract 1-simplex. However, if we also insist that the open sets corresponding to every intersection indexed by a set in N is also contractible, the situation changes. This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of N.

The Čech nerve[edit]

Given an open cover of a topological space , or more generally a cover in a site, we can regard the pairwise fibre products , which in the case of a topological space is precisely the intersection . 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, and by taking connected components we get a simplicial set, which we can realise topologically: . If the covering and the space are sufficiently nice, for instance if is compact and all intersections of sets in the cover are contractible or empty, then this space is weakly equivalent to . This is known as the nerve theorem.

See also[edit]


Notes[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. 

References[edit]

  • Artin, Michael; Mazur, Barry (1969). Etale homotopy. Springer. 
  • Nerve theorem in nLab
  • Samuel Eilenberg and Norman Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952, p. 234.