Constructing 𝑈𝑉^{𝑘}-maps between spheres

Ferry, Steven C.

doi:10.1090/S0002-9939-1994-1166355-5

Constructing $UV^ k$-maps between spheres
HTML articles powered by AMS MathViewer

by Steven C. Ferry PDF

Proc. Amer. Math. Soc. 120 (1994), 329-332 Request permission

Abstract:

The purpose of this note is to give a quick proof of an extremely counterintuitive theorem of Bestvina, Walsh, and Wilson. The theorem says, for example, that the degree $2$ map ${d_2}:{S^3} \to {S^3}$ is homotopic to a map such that ${p^{ - 1}}(x)$ is connected for each $x \in {S^3}$.

References

R. D. Anderson, Open mappings of compact continua, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 347–349. MR 78682, DOI 10.1073/pnas.42.6.347
Mladen Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988), no. 380, vi+110. MR 920964, DOI 10.1090/memo/0380

A generalization of Keldysh’s construction of a monotone map of a cube onto a cube of higher dimension

40

Marshall M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 0362320, DOI 10.1007/978-1-4684-9372-6
Steve Ferry, A stable converse to the Vietoris-Smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980), no. 2, 369–386. MR 580894, DOI 10.1090/S0002-9947-1980-0580894-1

Mapping manifolds to polyhedra

Lyudmila Keldyš, Monotone mapping of a cube onto a cube of higher dimension, Mat. Sb. N.S. 41(83) (1957), 129–158 (Russian). MR 0090805
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

A general method of constructing

mappings on manifolds with applications to spheres

Simplicial spaces, nucleii, and

groups

45

David C. Wilson, On constructing monotone and $UV^{1}$ mappings of arbitrary degree, Duke Math. J. 41 (1974), 103–109. MR 339250