Concerning the shapes of finite-dimensional compacta

Geoghegan, Ross; Summerhill, R. Richard

doi:10.1090/S0002-9947-1973-0324637-1

Concerning the shapes of finite-dimensional compacta
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by Ross Geoghegan and R. Richard Summerhill PDF

Trans. Amer. Math. Soc. 179 (1973), 281-292 Request permission

Abstract:

It is shown that two “tamely” embedded compacta of dimension $\leq k$ lying in ${E^n}(n \geq 2k + 2)$ have the same (Borsuk) shape if and only if their complements are homeomorphic. In particular, two k-dimensional closed submanifolds of ${E^{2k + 2}}$ have the same homotopy type if and only if their complements are homeomorphic.

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