Fundamental crossed square of a square of spaces

Question

I am a newbie at Topology and I didn't took the homotopy theory course. Sorry for that, but I need to know some facts about this paper ("Van Kampen Theorems for Diagrams of Spaces", authors R. Brown & J.-L. Loday, appearing in "Topology" vol. 26 No 3, pages 311-335, 1987)

(1) what are these fibres?

(2) how to prove the claim?

Thanks

Answer 1

Hi,

I think you need to work on your homotopy theory, particularly the notion of homotopy fibre. The following reference might be helpful, once you have done that!

Gilbert, N.D. "On the fundamental $\rm cat^ n$-group of an $n$-cube of spaces". In "Algebraic topology, Barcelona, 1986", Lecture Notes in Math., Volume 1298. Springer, Berlin (1987), 124--139.

You should compare this with the paper you reference as [20] and also with the Appendix to the paper you quote.

Some background to the related notion of triad groups is in J.F. Adams "Student's guide to algebraic topology".

The fact that the axioms for a crossed square are "right" is because of the equivalence of crossed squares with cat$^2$-groups, and that is non trivial, though the case of dimension $n$ has also been proved by G. Ellis and R. Steiner. JPAA 1987.

See also

R. Brown ``Computing homotopy types using crossed $n$-cubes of groups'', Adams Memorial Symposium on Algebraic Topology, Vol 1, edited N. Ray and G Walker, Cambridge University Press, 1992, 187-210.

which is available as [74] on my publication list.

There is on my preprint list pdf files of various presentations giving background.

This is not an easy area, but it is also one which not many know about, and there is surely a lot more that can be done. If you email me, I can give you some more notes on the basic algebra.

All this links with classical work on $n$-ad homotopy groups, but in some sense puts them all together and via the van Kampen theorem in the paper you quote allows for new nonabelian colimit calculations in homotopy theory. A bibiliography on the non abelian tensor product, a notion which arose from considering pushouts of crossed squares, has now 131 items, mainly by group theorists.