nLab
nerve theorem

This page is about a property of Cech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions. For the “nerve theorem” for monads with arities see there.

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Idea

The nerve theorem asserts that the homotopy type of a sufficiently nice topological space is encoded in the Cech nerve of a good cover.

This can be seen as a special case of some aspects of étale homotopy as the étale homotopy type of nice spaces coincides with the homotopy type of its Cech nerve.

Statement

Theorem

Let $X$ be a paracompact space and $\{U_i \to X\}$ a good open cover. Write $C(\{U_i\})$ for the Cech nerve of this cover

$$C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} U_i \cap U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i \right) \,,$$

(a simplicial space) and write

$$\tilde C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} * \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} * \right)$$

for the simplicial set obtained by replacing in $C(\{U_i\})$ each direct summand space by the point. Let $|\tilde C(\{U_i\})|$ be the geometric realization.

This is homotopy equivalent to $X$.

The proof relies on the existence of partitions of unity.

This is usually attributed to (Borsuk 1948). It appears more explicitly as Weil 52, p. 141 McCord 67, Thm. 2, review in Hatcher, prop. 4G.3.

Remark

This statement implies that in the cohesive (∞,1)-topos ETop∞Grpd the intrinsic fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos coincides with the ordinary fundamental ∞-groupoid functor of paracompact topological spaces. See Euclidean-topological ∞-groupoid : Geometric homotopy for details.

References

Original references:

• Karol Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math. 35, (1948) 217–234 (dml:213158)

• Jean Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue, J. Math. Pures Appl. (9) 29 (1950), 1–139.

• André Weil, §5. in: Sur les theoremes de de Rham, Comment. Math. Helv. 26 (1952), 119–145 (dml:139040)

• Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers, Proc. Amer. Math. Soc. 18 (1967), 705–708 (jstor:2035443)

• Graeme Segal, §4 in: Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.

• Armand Borel and Jean-Pierre Serre, Theorem 8.2.1. in: Corners and arithmetic groups,

  Comment. Math. Helv. 48 (1973), 436–491.

A version for hypercovers is discussed in

• Daniel Dugger, Daniel C. Isaksen, Topological hypercovers and $\mathbb{A}^1$-realizations, Math. Z. 246 (2004), no. 4, 667–689

A review appears as corollary 4G.3 in the textbook

• Allen Hatcher, Algebraic topology (web) .

Some slightly stronger statements are discussed in

• Anders Björner, Nerves, fibers and homotopy groups, Journal of combinatorial theory, series A, 102 (2003), 88-93

• Andrzej Nagórko, Carrier and nerve theorems in the extension theory, Proc. Amer. Math. Soc. 135 (2007), 551-558. (web)

A nerve theorem for categories:

• Kohei Tanaka, Cech complexes for covers of small categories, Homology, Homotopy and Applications 19(1), (2017), pp. 281-291. arXiv:1508.03688