An example of a fake 𝑠-manifold with a nice locally contractible compactification

Bowers, Philip L.

doi:10.1090/S0002-9939-1986-0848898-X

An example of a fake $s$-manifold with a nice locally contractible compactification
HTML articles powered by AMS MathViewer

by Philip L. Bowers PDF

Proc. Amer. Math. Soc. 98 (1986), 171-174 Request permission

Abstract:

An example is constructed of a topologically complete separable AR $X$ that satisfies the discrete $n$-cells property for each nonnegative integer $n$ but fails to satisfy the discrete approximation property and be homeomorphic to $s$ even though $X$ arises as the complement of a $\sigma$-$Z$-set in a locally contractible compactum. Such examples are not possible in the setting of ANR compactifications.

References

Mladen Bestvina, Philip Bowers, Jerzy Mogilski, and John Walsh, Characterization of Hilbert space manifolds revisited, Topology Appl. 24 (1986), no. 1-3, 53–69. Special volume in honor of R. H. Bing (1914–1986). MR 872478, DOI 10.1016/0166-8641(86)90049-0
Karol Borsuk, Sur un espace compact localement contractile qui n’est pas un rétracte absolu de voisinage, Fund. Math. 35 (1948), 175–180 (French). MR 27498, DOI 10.4064/fm-35-1-175-180
Philip L. Bowers, Discrete cells properties in the boundary set setting, Proc. Amer. Math. Soc. 93 (1985), no. 4, 735–740. MR 776212, DOI 10.1090/S0002-9939-1985-0776212-6

Nonshrinkable ’cell-like’ decompositions of

Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
D. M. Hyman, A generalization of the Borsuk-Whitehead-Hanner theorem, Pacific J. Math. 23 (1967), 263–271. MR 225295, DOI 10.2140/pjm.1967.23.263
A. H. Kruse and P. W. Liebnitz, An application of a family homotopy extension theorem to $\textrm {ANR}\$ spaces, Pacific J. Math. 16 (1966), 331–336. MR 195091, DOI 10.2140/pjm.1966.16.331
H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 10.4064/fm-111-3-247-262