Certain continua in 𝑆ⁿ of the same shape have homeomorphic complements

Liem, Vo Thanh

doi:10.1090/S0002-9947-1976-0397737-0

Certain continua in $S^{n}$ of the same shape have homeomorphic complements
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Trans. Amer. Math. Soc. 218 (1976), 207-217 Request permission

Abstract:

As a consequence of Theorem 1 of this paper, we see that if X and Y are globally 1-alg continua in ${S^n}\;(n \geqslant 5)$ having the shape of the real projective space ${P^k}\;(k \ne 2,2k + 2 \leqslant n)$, then ${S^n} - X \approx {S^n} - Y$. (For ${P^1} = {S^1}$, this establishes the last case of such a result for spheres.) We also show that if X and Y are globally 1-alg continua in ${S^n},n \geqslant 6$, which have the shape of a codimension $\geqslant 3$, closed, $0 < (2m - n + 1)$-connected, PL-manifold ${M^m}$, then ${S^n} - X \approx {S^n} - Y$.

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