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
HTML articles powered by AMS MathViewer

by Vo Thanh Liem

Trans. Amer. Math. Soc. 218 (1976), 207-217

DOI: https://doi.org/10.1090/S0002-9947-1976-0397737-0

PDF | 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 ${S^n} - X \approx {S^n} - Y$.

References

Karol Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223–254. MR 229237, DOI 10.4064/fm-62-3-223-254
William Browder, Structures on $M\times R$, Proc. Cambridge Philos. Soc. 61 (1965), 337–345. MR 175136
T. A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972), no. 3, 181–193. MR 320997, DOI 10.4064/fm-76-3-181-193
T. A. Chapman, Shapes of finite-dimensional compacta, Fund. Math. 76 (1972), no. 3, 261–276. MR 320998, DOI 10.4064/fm-76-3-261-276

A small loop condition for embedded compacta

D. Coram and P. F. Duvall Jr., Neighborhoods of sphere-like continua, General Topology and Appl. 6 (1976), no. 2, 191–198. MR 400238, DOI 10.1016/0016-660X(76)90032-5
Robert J. Daverman, Homotopy classification of complements of locally flat codimension two spheres, Amer. J. Math. 98 (1976), no. 2, 367–374. MR 420632, DOI 10.2307/2373890
Robert J. Daverman, On weakly flat $1$-spheres, Proc. Amer. Math. Soc. 38 (1973), 207–210. MR 310894, DOI 10.1090/S0002-9939-1973-0310894-X
Paul F. Duvall Jr., Weakly flat spheres, Michigan Math. J. 16 (1969), 117–124. MR 246300
Ross Geoghegan and R. Richard Summerhill, Concerning the shapes of finite-dimensional compacta, Trans. Amer. Math. Soc. 179 (1973), 281–292. MR 324637, DOI 10.1090/S0002-9947-1973-0324637-1
V. K. A. M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres. I, II, Proc. London Math. Soc. (3) 3 (1953), 29–53, 129–152. MR 58204, DOI 10.1112/plms/s3-3.1.29
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
M. C. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965), 1–14. MR 182978, DOI 10.2307/1970560
Michel A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105–119. MR 0178475
Sibe Mardešić and Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), no. 1, 41–59. MR 298634, DOI 10.4064/fm-72-1-41-59
T. B. Rushing, The compacta $X$ in $S^{n}$ for which $\textrm {Sh}(X)=\textrm {Sh}(S^{k})$ is equivalent to $S^{n}-X\approx S^{n}-S^{k}$, Fund. Math. 97 (1977), no. 1, 1–8. MR 461515, DOI 10.4064/fm-97-2-79-94

The obstruction to finding a boundary for an open manifold of dimension greater than five

L. C. Siebenmann, On detecting open collars, Trans. Amer. Math. Soc. 142 (1969), 201–227. MR 246301, DOI 10.1090/S0002-9947-1969-0246301-9
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112

Seminar on combinatorial topology

C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
Sibe Mardešić and Jack Segal, Equivalence of the Borsuk and the ANR-system approach to shapes, Fund. Math. 72 (1971), no. 1, 61–68. (errata insert). MR 301694, DOI 10.4064/fm-72-1-61-68