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?