Rigid 3-dimensional compacta whose squares are manifolds

Ancel, Fredric D.; Duvall, Paul F.; Singh, S.

doi:10.1090/S0002-9939-1983-0695269-2

Rigid $3$-dimensional compacta whose squares are manifolds
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by Fredric D. Ancel, Paul F. Duvall and S. Singh PDF

Proc. Amer. Math. Soc. 88 (1983), 330-332 Request permission

Abstract:

A space is rigid if its only self-homeomorphism is the identity. In response to a question of Jan van Mill, Ancel and Singh have given examples of rigid $n$-dimensional compacta, for each $n \geqslant 4$, whose squares are manifolds. We construct a rigid $3$-dimensional compactum whose square is the manifold ${S^3} \times {S^3}$. In fact, we construct uncountably many topologically distinct compacta with these properties.

References

Fredric D. Ancel and S. Singh, Rigid finite-dimensional compacta whose squares are manifolds, Proc. Amer. Math. Soc. 87 (1983), no. 2, 342–346. MR 681845, DOI 10.1090/S0002-9939-1983-0681845-X
A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
Charles D. Bass, Some products of topological spaces which are manifolds, Proc. Amer. Math. Soc. 81 (1981), no. 4, 641–646. MR 601746, DOI 10.1090/S0002-9939-1981-0601746-0
E. M. Brown, Contractible $3$-manifolds of finite genus at infinity, Trans. Amer. Math. Soc. 245 (1978), 503–514. MR 511426, DOI 10.1090/S0002-9947-1978-0511426-2
R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), no. 4, 495–552. MR 645403, DOI 10.1090/S0002-9904-1977-14321-8
D. R. McMillan Jr., Some contractible open $3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373–382. MR 137105, DOI 10.1090/S0002-9947-1962-0137105-X
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), no. 3, 597–600. MR 627701, DOI 10.1090/S0002-9939-1981-0627701-2