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^[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 definition

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

a finite set J ⊆ I belongs to N if and only if the intersection of the U_i whose subindices are in J is non-empty,

\bigcap_{j\in J}U_j \neq \varnothing.

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 are also contractible, the situation changes. In particular, the nerve lemma states that if ${\mathcal {U}}=\{U_{i}\}_{i\in I}$ is a good cover, in that for each $\sigma \subset I$ , the set $\bigcap _{i\in \sigma }U_{i}$ is contractible if it is nonempty, then the nerve $N({\mathcal {U}})$ is homotopy equivalent to $\bigcup _{i\in I}U_{i}$ . ^[2]

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

Given an open cover ${\mathcal {U}}=\{U_{i}\}$ of a topological space $X$ , or more generally a cover in a site, we can regard the pairwise fibre products $U_{ij}=U_{i}\times _{X}U_{j}$ , which in the case of a topological space is precisely the intersection $U_{i}\cap U_{j}$ . The collection of all such intersections can be referred to as ${\mathcal {U}}\times _{X}{\mathcal {U}}$ and the triple intersections as ${\mathcal {U}}\times _{X}{\mathcal {U}}\times _{X}{\mathcal {U}}$ . By considering the natural maps $U_{ij}\to U_{i}$ and $U_{i}\to U_{ii}$ , we can construct a simplicial object $C({\mathcal {U}})_{\bullet }$ defined by $C({\mathcal {U}})_{n}={\mathcal {U}}\times _{X}\cdots \times _{X}{\mathcal {U}}$ , n-fold fibre product. This is the Čech nerve, and by taking connected components we get a simplicial set, which we can realise topologically: $|C(\pi _{0}({\mathcal {U}}))|$ . If the covering and the space are sufficiently nice, for instance if $X$ is compact and all intersections of sets in the cover are contractible or empty, then this space is weakly equivalent to $X$ . This is known as the nerve theorem.

Notes

↑ Aleksandroff, P. S. (1928). "Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung". Mathematische Annalen. 98: 617–635. doi:10.1007/BF01451612.
↑ 1969-, Ghrist, Robert W.,. Elementary applied topology (Edition 1.0 ed.). [United States]. ISBN 9781502880857. OCLC 899283974.

References

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.

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[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.

[2] 1969-, Ghrist, Robert W.,. Elementary applied topology (Edition 1.0 ed.). [United States]. ISBN 9781502880857. OCLC 899283974.

Nerve of a covering

Alexandrov's definition

The Čech nerve

See also

Notes

References