(Higher Homotopy Seifert-van Kampen Theorems for diagrams of spaces)

There is a certain philosophy and approach which has to be understood to see what is going on for these theorems in which I have been involved.

It is standard to talk about "the fundamental group of a space", but this is strictly speaking a misconception: we are really allowed only to speak of "the fundamental group of a pointed space". Similarly, we can speak only of "the universal cover of a pointed space". Grothendieck has written in aletter to RB in 1983:

"-both the choice of a base point, and the 0-connectedness assumption, however innocuous they may seem at first sight, seem to me of a very essential nature. To make an analogy, it would be just impossible to work at ease with algebraic varieties, say, if sticking from the outset (as had been customary for a long time) to varieties which are supposed to be connected. Fixing one point, in this respect (which wouldn't have occurred in the context of algebraic geometry) looks still worse, as far as limiting elbow-freedom goes! Also, expressing a pointed 0-connected homotopy type in terms of a group object mimicking the loop space (which isn't a group object strictly speaking), or conversely, interpreting the group object in terms of a pointed "classifying space", is a very inspiring magic indeed - what makes it so inspiring it that it relates objects which are definitively of a very different nature - let's say, "spaces" and "spaces with group law". The magic shouldn't make us forget though in the end that the objects thus related are of different nature, and cannot be confused without causing serious trouble."

The problem with the usual single base point approach to the Seifert-van Kampen Theorem (SvKT) is that it does not compute the, or any, fundamental group of the circle. This shows how embarrassing is, or should be, the restriction to path-connected spaces. Further a solution was published in 1967, namely to used the fundamental groupoid on a set of base points, chosen according to the geometry. In particular, one represents the cirle C as the union of two connected open sets whose intersection has two path components. It is therefore convenient to choose two base points, say +1 and -1, one in each component. One then computes π₁(C,{+1.-1}).

To use this kind of result, one has to develop some of the algebra of groupoids. One basic technique is to consturct a groupoid f_*G from a groupoid G and a function f from Ob(G) to a set Y. This construction, made using equivalence classes of words, in effect includes the construction of free groups, free products of groups and the important free groupoids. This, and choices of trees, enables one to compute in practical terms pushouls and colimis of groupoids.

Essentially, the SvKT with sets of base points enables computation of homotopy 1-types. This is possible because groupoids have structure in dimensions 0 and 1, and so can model the information needed for gluing 1-types.

It is therefore reasonable to search for algebraic structures which have structure in dimensions 0,...,n, and a corresponding SvKT.

Also it seemed to me in 1965 that some aspects of the proof of the groupoid version of the SvKT generalised to dimension 2 if one had the right "homotopy gadget": the main features would be composition of squares and a notion of commutative cube, such that any composition of commutative cubes is commutative.

A good way forward for the dimension 2 problem was found in discussion with Philip Higgins in 1994, by making a reqirement that we should be able to recover a subtle theorem of JHC Whitehead on free crossed modules (1942-49), a theorem that is sometimes quoted but rarely proved in algebraic topology texts. This theorem was on relative homotopy groups, as crossed modules. It therefore seemed logical to start with a relative situation, i.e. a space X with a subspace A and a base point, c, say. Then one considers homotopy classes rel vertices of maps of a 2-cube I² into X which takes the edges into A and the vertices to c. We quickly found that this gave a 2-dimensional SvKT with a great generalisation of Whitehead's theorem as a consequence! It showed essentially that one can compute homotopy 2-types of some colimits of spaces as colimits of crossed modules. Note that a related theorem in terms of k-invariants seems not possible. Indeed, one has to "dig out" from such a colimit of crossed modules information on the 2nd homotopy group. This is quite analogous to the 1-dimensional situation with groupoids as a model of homotopy 1-types.

The paper was submitted in 1975; it aroused stiff opposition, without explanation. A cut down version was finally accepted, and published in 1978, in Proc. LMS.

With this experience of development of the 2-dimensional case, and related to further work of J.H.C. Whitehead in his not so well studied paper "Combinatoriual Homotopy II", a version in all dimensions was worked out using filtered spaces, and published in JPAA in 1981. This work is covered in the book Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy gropoids . A central feature of this book is the use of cubical methods, both to prove a Higher Homotopy Seifert-van Kampen Thorem, and to develop monoidal closed category structures. For some background to this development, see this 2014 seminar A homotopical approach to algebraic topology via compositions of cubes .

Further developments to (pointed) homotopy theory were made in papers with J.-L. Loday published in 1987. They relate to the longstanding tradition in homotopy theory of using triadic and (n+1)-adic homotopy groups, but now generalised to n-cubes of spaces. Loday defined a functor Π from n-cubes of pointed spaces to what we later called catⁿ-groups. It was this functor we proved satisfied a higher homotopy Seifert-van Kampen Theorem, of the following form.

Theorem
Let X be an n-cube of pointed spaces. Let {U_λ}_{λ ∈ Λ} be an open covering of X(1, ..., 1), so that each U_σ for σ∈ Λ_fin determines by inverse image an n-cube of spaces U_σ. Suppose that each such U_σ is a connected n-cube. Then the following hold:
(Con): the n-cube X is connected, and
(Iso): the following natural homomorphism of catⁿ-groups is an isomorphism:

colimσcat ΠUσ→ ΠX = Πcolimσ Uσ.

Notice in this theorem the key role of the notion of a connected n-cube of spaces. This means that when the n-cube of spaces is converted to an n-cube of fibrations, then all the spaces in the latter are connected spaces.

This theorem and related algebra have been applied by G.J. Ellis and R. Steiner to give a solution to an old problem, that of the critical group in the n-ad connectivity theorem of Barratt and Whitehead.

Note that there is considerable work on using geometric methods to prove the n-ad connectivity theorem, but these methods do not reach the algebraic results, which in fact imply the major part of the connectivity results.

There are interesting results also on excision in the paper with Loday on "Homotopical excision and HUrewicz Theorems for n-cubes of spaces. The initial problem was a conjecture of Loday which we identified as a conjectured triadic Hurewicz Theorem. Following the scheme of my work with Higgins, this should follow from a triadic van Kampen theorem. It did, but needed a further analysis of: "What should be excision when a space is the union of n subsets? This turned out to be an n-cube of (n - 1)-cubes?"! This method gives new algebraic results, such as an n-adic Hurewicz Theorem, and even the connectivity deductions may not have been obtained by more geometric methods. (I have asked about this,)

Updated 27 April, 2016.

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