On the shape of torus-like continua and compact connected topological groups
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- by James Keesling PDF
- Proc. Amer. Math. Soc. 40 (1973), 297-302 Request permission
Abstract:
In this paper it is shown that if $X$ is a torus-like continuum, then $X$ has the shape of a compact connected abelian topological group. Let $\Pi$ be a collection of compact connected Lie groups. In light of the above result it is natural to ask if a $\Pi$-like continuum has the shape of a compact connected topological group. An example is given to show that this is not the case.References
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144 K. H. Hofmann, Introduction to the theory of compact groups, Part II, Tulane University Lecture Notes, Tulane University, 1968.
- Karl Heinrich Hofmann, Categories with convergence, exponential functors, and the cohomology of compact abelian groups, Math. Z. 104 (1968), 106–140. MR 228615, DOI 10.1007/BF01109874
- Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
- W. Holsztyński, An extension and axiomatic characterization of Borsuk’s theory of shape, Fund. Math. 70 (1971), no. 2, 157–168. MR 282368, DOI 10.4064/fm-70-2-157-168 J. Keesling, Shape theory and compact connected abelian topological groups (submitted).
- James Keesling, Continuous functions induced by shape morphisms, Proc. Amer. Math. Soc. 41 (1973), 315–320. MR 334141, DOI 10.1090/S0002-9939-1973-0334141-8 —, An algebraic property of the Čech cohomology groups which prevents local connectivity and movability (submitted).
- Sibe Mardešić, $cyeps$-mappings and inverse limits, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 18 (1963), 195–205 (English, with Serbo-Croatian summary). MR 165501
- Sibe Mardešić and Jack Segal, $\varepsilon$-mappings onto polyhedra, Trans. Amer. Math. Soc. 109 (1963), 146–164. MR 158367, DOI 10.1090/S0002-9947-1963-0158367-X
- S. Mardešić and J. Segal, Movable compacta and $\textrm {ANR}$-systems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 649–654 (English, with Russian summary). MR 283796
- 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
- L. S. Pontriaguin, Continuous groups, Editorial Mir, Moscow, 1978 (Spanish). Translated from the third Russian edition by Carlos Vega. MR 528130
- Wladimiro Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562–567. MR 301130, DOI 10.1090/S0002-9939-1972-0301130-8
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 297-302
- MSC: Primary 54C56; Secondary 22C05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319140-4
- MathSciNet review: 0319140