Duality between proper homotopy theory and strong shape theory - MathOverflow

Duality between proper homotopy theory and strong shape theory

2011-07-21T08:09:05Z

In the n-lab entry about shape theory one can read that

Strong Shape Theory [...] has, especially in the approach pioneered by Edwards and Hastings, strong links to proper homotopy theory. The links are a form of duality related to some of the more geometric duality theorems of classical cohomology.

I would be interested in any reference where I can find a precise formulation of this duality.

EDIT: according to Gjergji Zaimi's answer the duality might be an improvement of Chapman's complement theorem. One can find it as Theorem 6.5.3 on page 230 of the book by Edwards and Hastings ("Cech and Steenrod Homotopy Theories with Applications to Geometric Topology"). Nevertheless, it seems to me that what was meant on the n-lab entry was more a cohomology type duality (like an instance of Verdier duality in the $(\infty,1)$-ctageorical context). Am I completely wrong?

Answer by Gjergji Zaimi for Duality between proper homotopy theory and strong shape theory

2011-07-21T08:39:02Z

A quick summary of the story is told in section 7 of S. Mardešić's "Shape Theory" from the ICM proceedings (1978) (find here). Strong shape theory was introduced by Edwards and Hastings keeping in mind this duality, which was inspired by Chapman's complement theorem (mentioned in the nlab article).

Answer by Tim Porter for Duality between proper homotopy theory and strong shape theory

2011-12-13T17:08:41Z

I wrote that part of the nLab entry so can confirm that it is the Edwards and Hastings extension of Chapman's result that was referred to, but my feeling in this is that that result is the geometric form of a lot of the classical cohomological duality results and that there should be more to be said about this ... but I don't know what! Perhaps looking at the Chapman result in the light of modern homotopy theory (say using Lurie's notion of shape) may give an $(\infty,1)$-categorical result. (Note that Batanin did work on strong shape theory and produced an $A_\infty$-structure, which must relate to this. Now I like that set of ideas. Good luck if you try it!)

(You may spur me on to write more on that entry as it has got stalled... Alternatively anyone else is welcome to write more on strong shape, of course.... and to correct any miswording, typos that they find. :-))