Nerve of a covering

Let X be a topological space. Let ${\displaystyle I}$ be an index set. Let ${\displaystyle C}$ be a family indexed by ${\displaystyle I}$ of open subsets of X: ${\displaystyle C=\lbrace U_{i}:i\in I\rbrace }$. The nerve of ${\displaystyle C}$ is a set of finite subsets of the index-set ${\displaystyle I}$. It contains all finite subsets ${\displaystyle J\subseteq I}$ such that the intersection of the U_i whose subindices are in J is non-empty:

N(C) := ${\displaystyle {\bigg \{}J:J\subseteq I~~~{\text{and}}~~~\bigcap _{j\in J}U_{j}\neq \varnothing {\bigg \}}}$

N(C) may contain singletons (elements i in ${\displaystyle I}$ such that U_i is non-empty), pairs (pairs of elements of i,j in ${\displaystyle I}$ such that U_i intersects U_j), triplets, and so on. If J belongs to N(C), then any of its subsets is also in N(C). Therefore N(C) is an abstract simplicial complex and it is often called the nerve complex of C.

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 U_i is an arc covering one third of S¹, with some overlap with the adjacent U_i. 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.

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

By considering the natural maps ${\displaystyle U_{ij}\to U_{i}}$ and ${\displaystyle U_{i}\to U_{ii}}$, we can construct a simplicial object ${\displaystyle S(C)_{\bullet }}$ defined by ${\displaystyle S(C)_{n}=C\times _{X}\cdots \times _{X}C}$, n-fold fibre product. This is the Čech nerve. [3]

By taking connected components we get a simplicial set, which we can realise topologically: ${\displaystyle |S(\pi _{0}(C))|}$.

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 ${\displaystyle J\subset I}$ the set ${\displaystyle \bigcap _{i\in J}U_{i}}$ 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 ${\displaystyle X}$ is compact and all intersections of sets in C are contractible or empty, then the space ${\displaystyle |S(\pi _{0}(C))|}$ is homotopy-equivalent to ${\displaystyle X}$.[6]

• Hypercovering