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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the nerve complex of a
set family In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set f ...
is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by
Pavel Alexandrov Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
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.


Basic definition

Let I be a set of indices and C be a family of sets (U_i)_. 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:'', Section 4.3'' :N(C) := \bigg\. In Alexandrov's original definition, the sets (U_i)_ are
open subsets In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
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 \neq \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 In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
. Hence N(C) is often called the nerve complex of C.


Examples

# Let ''X'' be the circle S^1 and C = \, 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) = \, which is an abstract 1-simplex. # Let ''X'' be the circle S^1 and C = \, where each U_i is an arc covering one third of S^1, with some overlap with the adjacent U_i. Then N(C) = \. 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

Given an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
C=\ of a topological space X, or more generally a cover in a
site Site most often refers to: * Archaeological site * Campsite, a place used for overnight stay in an outdoor area * Construction site * Location, a point or an area on the Earth's surface or elsewhere * Website, a set of related web pages, typically ...
, we can consider the pairwise fibre products U_=U_i\times_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\times_X C and the triple intersections as C\times_X C\times_X C. By considering the natural maps U_\to U_i and U_i\to U_, we can construct a
simplicial object In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
S(C)_\bullet defined by S(C)_n=C\times_X\cdots\times_XC, n-fold fibre product. This is the Čech nerve. By taking connected components we get a
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
, 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 \bigcup 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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to the original circle. 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 ''\bigcup C''.


Leray's nerve theorem

The basic nerve theorem of
Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
says that, if any intersection of sets in N(C) is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
(equivalently: for each finite J\subset I the set \bigcap_ U_i is either empty or contractible; equivalently: ''C'' is a good open cover), then N(C) is
homotopy-equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to ''\bigcup C''.


Borsuk's nerve theorem

There is a discrete version, which is attributed to Borsuk.'''' 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 by ''N''. If, for each nonempty J\subset I, the intersection \bigcap_ U_i is either empty or contractible, then ''N'' is
homotopy-equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to ''K''. A stronger theorem was proved by Anders Bjorner. if, for each nonempty J\subset I, the intersection \bigcap_ U_i is either empty or (k-, J, +1)-connected, then for every ''j'' ≤ ''k'', the ''j''-th
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of ''N'' is isomorphic to the ''j''-th
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to X.


Homological nerve theorem

The following nerve theorem uses the
homology groups In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of intersections of sets in the cover. For each finite J\subset I, denote H_ := \tilde_j(\bigcap_ U_i)= the ''j''-th
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
group of \bigcap_ U_i. If ''HJ,j'' is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
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_j(N(C)) \cong \tilde_j(X) for all ''j'' in ; * if \tilde_(N(C))\not\cong 0 then \tilde_(X)\not\cong 0 .


See also

*
Hypercovering In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every in ...


References

{{DEFAULTSORT:Nerve Of A Covering Topology Simplicial sets Families of sets