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higher homotopy van Kampen theorem

Contents

Idea

A higher homotopy van Kampen theorem is a theorem that asserts that the homotopy type of a topological space can be computed by a suitable colimit or homotopy colimit over homotopy types of its pieces.

This generalizes the van Kampen theorem, which only deals with the underlying 1-type (the fundamental groupoid).

Statement

For topological spaces

General

Theorem

Let XX be a topological space, write Op(X)Op(X) for its category of open subsets and let

χ:COp(X) \chi : C \to Op(X)

be a functor out of a small category CC such that

Then:

the canonical morphism in sSet out of the colimit

lim SingχSing(X) {\lim_\to} Sing \circ \chi \to Sing(X)

into the singular simplicial complex of XX exhibits Sing(X)Sing(X) as the homotopy colimit hocolimSingχhocolim Sing \circ \chi.

This is theorem A.1.1 in (Lurie).

Strict version

The following is a version of the above general statement restricted to a strict ∞-groupoid-version of the fundamental ∞-groupoid and applicable for topological spaces that are equipped with the extra structure of a filtered topological space.

Notice that these strict \infty-groupoids are equivalent to crossed complexes.

Suppose X *X_* is a filtered space and XX is the union of the interiors of sets U iU^i, iIi \in I. Let U * iU^i_* be the filtered space given by the intersections U iX nU^i \cap X_n for n0n \geq 0. If d=(i,j)I 2d=(i,j) \in I^2 we write U dU^d for U iU jU^i \cap U^j. We then have a coequaliser diagram of filtered spaces

dI 2U * d b a iIU * i cX *.\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.
Strict Higher van Kampen Theorem

If the filtered spaces U * fU^f_* are connected filtered spaces for all finite intersections U * fU^f_* of the filtered spaces U * iU^i_*, then

  1. (Conn) The filtered space X *X_* is connected; and

  2. (Iso) The fundamental crossed complex functor Π\Pi takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.

Remarks

  • Note that because Π\Pi uses groupoids, it obviously takes disjoint unions \bigsqcup of filtered spaces into disjoint unions (= coproducts) \bigsqcup of crossed complexes.

  • The proof of the theorem is not direct but goes via the fundamental cubical ω\omega-groupoid with connections of the filtered spaces, as that context allows the notions of algebraic inverse to subdivision and of commutative cube. However the proof is a direct generalisation of a proof for the van Kampen theorem for the fundamental groupoid.

  • Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of singular homology theory or of simplicial approximation.

For objects in a cohesive (,1)(\infty,1)-topos

In a cohesive (∞,1)-topos (already in a locally ∞-connected (∞,1)-topos) higher van Kampen theorems hold in great generality.

See the section cohesive (∞,1)-topos – van Kampen theorem.

In particular for the cohesive (,1)(\infty,1)-topos ∞TopGrpd of topological ∞-groupoids this reproduces the topological higher van Kampen theorem discussed above.

Examples

Here is one application in dimension 2 not easily obtainable by traditional algebraic topology.

Let 0PQR00 \to P \to Q \to R \to 0 be an exact sequence of abelian groups. Let XX be the mapping cone of the induced map K(P,1)K(Q,1)K(P,1) \to K(Q,1) of Eilenberg-Mac Lane spaces. Then a crossed module representing the homotopy 2-type of XX is μ:CQ\mu: C \to Q where CC is abelian and is the direct sum rRP r\oplus_{r \in R} P^r of copies of PP one for each rRr \in R and the action of QQ is via RR and permutes the copies by (p,r) s=(p,r+s)(p,r)^s=(p,r+s). Similar examples for P,Q,RP,Q,R nonabelian are do-able, more complicated, and certainly not obtainable by traditional methods.

References

The version for topological spaces and the fundamental infinity-groupoid functor is discussed in Appendix A of

The version for filtered topological spaces and the strict homotopy \infty-groupoid functor is discussed in

Revised on May 5, 2011 08:37:46 by Urs Schreiber (89.204.137.24)