A higher homotopy van Kampen theorem is a theorem that asserts that the homotopy type, or certain invariants thereof, of a topological space can be ‘computed’, in some sense, by a suitable colimit or homotopy colimit over the homotopy types / invariants of its pieces.
These all in some sense generalize the van Kampen theorem, (SvKT), which only deals with the underlying 1-type (the fundamental groupoid). The actual statements of the results depend on the models of the homotopy invariants being considered.
The general form that this takes is similar to the classical homology small simplex theorem which can be found in many books on homology. This states that given an open cover, , of a space, , the inclusion of the chain complex of -small singular simplices into the singular complex of induces isomorphisms on all homology groups. It is closely related to results on excision and on the Mayer-Vietoris theorem. Various homotopy analogues of this result have been given with one of the most recent being that given by Luire (see below). He explicitly relates it to the van Kampen theorem, and it is a similar ‘local-to-global’ result.
The experience from the uses of the homology small simplex theorem suggests that it may be most useful when used in conjunction with specific invariants of homotopy types. (Note excision fails for homotopy groups and so one has to be more subtle.) The precise relationship to the other forms of generalised van Kampen theorem has yet to be determined.
This is theorem A.1.1 in (Lurie).
This form of the general type of result gives a relationship between the homotopy type of the space and the homotopy types of parts of that space, telling one how they fit together. To get explicit more ‘algebraic’ calculations out it has so far proved necessary to consider spaces with extra structure, such as a filtration (Brown-Higgins). The explicit algebraic models do not, in general, give a complete classifying invariant of the homotopy types concerned, but that allows much more precise and calculable invariants to be found. In the first ‘strict’ version, the invariant, namely the fundamental ∞-groupoid, is a strict ∞-groupoid and could be substituted by the corresponding crossed complex (according to your taste and needs!)
Suppose is a filtered space and is the union of the interiors of sets , . Let be the filtered space given by the intersections for . If we write for . We then have a coequaliser diagram of filtered spaces
If the filtered spaces are connected filtered spaces for all finite intersections of the filtered spaces , then
(Conn) The filtered space is connected; and
(Iso) The fundamental crossed complex functor takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.
Note that because uses groupoids, it obviously takes disjoint unions of filtered spaces into disjoint unions (= coproducts) of crossed complexes.
The proof of the theorem is not direct but goes via the fundamental cubical -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.
This result includes the crossed module version of the generalised SvKT. This was the first extension in this direction. It handles relative homotopy groups. That version is also a special case of the next of this type of result given by the Brown-Loday theory.
Artin and Mazur proved the existence of a spectral sequence determined by the Cech complex of an open cover. This converges to the homotopy groups of the big space. There does nnot seem to have been any study done of the relationship of their results with the others mentioned above. Their methods are very similar to those used for the Dugger-Isaksen small simplex theorem, however.
In a cohesive (∞,1)-topos higher van Kampen theorems hold in great generality.
See the section cohesive (∞,1)-topos – van Kampen theorem.
Here is one application in dimension 2 not easily obtainable by traditional algebraic topology.
Let be an exact sequence of abelian groups. Let be the mapping cone of the induced map of Eilenberg-Mac Lane spaces. Then a crossed module representing the homotopy 2-type of is where is abelian and is the direct sum of copies of one for each and the action of is via and permutes the copies by . Similar examples for nonabelian are do-able, more complicated, and certainly not obtainable by traditional methods.
Various versions exist, generalising the classical theorem.
The homology small simplex theorem is to be found, for instance, in Spanier’s book on algebraic topology.
A very similar result occurs before as prop. A.5 in
An important central lemma (the small simplices theorem) also appears as corollary 3.5 of
where many similar results are discussed.
The version for filtered topological spaces and the strict homotopy -groupoid functor was one of the aims of the long research programme of Ronnie Brown and Philip Higgins. For the original work see the publication lists of Brown and Higgins. A complete view of this approach is discussed in
The Brown-Loday results can be found in
The paper of Artin and Mazur is