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Rigid finite-dimensional compacta whose squares are manifolds


Authors: Fredric D. Ancel and S. Singh
Journal: Proc. Amer. Math. Soc. 87 (1983), 342-346
MSC: Primary 54G20; Secondary 55M15, 57P99
DOI: https://doi.org/10.1090/S0002-9939-1983-0681845-X
MathSciNet review: 681845
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Abstract: A space is rigid if its only self-homeomorphism is the identity. We answer questions of Jan van Mill by constructing for each $ n$, $ 4 \leqslant n < \infty $, a rigid $ n$-dimensional compactum whose square is homogeneous because it is a manifold. Moreover, for each $ n$, $ 4 \leqslant n < \infty $, we give uncountably many topologically distinct such examples. Infinite-dimensional examples are also given.


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  • [A] A. V. Arhangel'skiĭ, Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), 33-96. MR 526012 (80i:54005)
  • [B] C. D. Bass, Some products of topological spaces which are manifolds, Proc. Amer. Math. Soc. 81 (1981), 641-646. MR 601746 (82a:57012)
  • [DS] R. J. Daverman and S. Singh, Arcs in the Hilbert cube $ ({S^n})$ whose complements have different fundamental groups, Compositio Math. (to appear) MR 700004 (84h:57008)
  • [K] M. A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67-72. MR 0253347 (40:6562)
  • [L] R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 495-552. MR 0645403 (58:31095)
  • [M] W. S. Massey, Algebraic topology: An introduction, Harcourt Brace and World, New York, 1967. MR 0211390 (35:2271)
  • [Ma] B. Mazur, A note on some contractible $ 4$-manifolds, Ann. of Math. 73 (1961), 221-228. MR 0125574 (23:A2873)
  • [S] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton University, 1965.
  • [Sm] S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391-406. MR 0137124 (25:580)
  • [T] H. Toŕunczyk, On CE images of the Hilbert cube and characterization of $ Q$-manifolds, Fund. Math. 106 (1980), 31-40. MR 585543 (83g:57006)
  • [vM] Jan van Mill, A rigid space $ X$ for which $ X \times X$ is homogeneous; an application of infinite-dimensional topology, Proc. Amer. Math. Soc. 83 (1981), 597-600. MR 627701 (82h:54067)
  • [W] R. L. Wilder, Monotone mappings of manifolds, Pacific J. Math. 7 (1957), 1519-1528. MR 0092966 (19:1188e)
  • [Z] E. C. Zeeman, On the dunce hat, Topology 2 (1964), 341-358. MR 0156351 (27:6275)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0681845-X
Keywords: Generalized $ n$-manifold, decomposition spaces, homology spheres, homogeneous, cell-like decomposition
Article copyright: © Copyright 1983 American Mathematical Society

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