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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?

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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).

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  • $\begingroup$ Thank you very much. The statement is then (as far as I understand) Theorem 6.5.3 in on page 230 of the nook by Edwards and Hastings. I stupidly thought the duality they were refering too in the n-lab entry was an incarnation of Verdier duality. $\endgroup$
    – DamienC
    Commented Jul 21, 2011 at 9:09
  • $\begingroup$ A bit more on the history: Strong shape was introduced in a 2-truncated form by Christie in 1945. It was used by myself in Math. Zeit. 150, 1974, pp. 1-21, and I think more or less simultaneously by Mardesic and by, of course, Edwards and Hastings. The idea of the duality is already there in Sitnikov's work (but I do not have the precise reference to hand.) $\endgroup$
    – Tim Porter
    Commented Jan 19, 2014 at 6:41
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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. :-))

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  • $\begingroup$ Many thanks for your answer. I am a total beginner with shape theory (and not really an expert in homotopy theory). I guess the paper of Batanin you refer to is web.science.mq.edu.au/~mbatanin/SHAPE0.dvi I'll have a look at it. Tanks again. $\endgroup$
    – DamienC
    Commented Dec 13, 2011 at 20:32
  • $\begingroup$ Damien: the cohomological duality inherent in the Edwards and Hastings more precise version of Chapman's result is the geometric side of the very classical duality theorems of algebraic topology. The algebraic geometry side of things is less immediately `geometric' as it deals with sheaves. Perhaps one way to see things is to think of proper homotopy theory as being about exhausting a space by compact bits, (and so about the Ind-category of spaces), whilst shape is about the Pro-category. These categories also arise in Verdier's derived functors so ... ??? $\endgroup$
    – Tim Porter
    Commented Jan 19, 2014 at 6:36

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