Continuous functions induced by shape morphisms
HTML articles powered by AMS MathViewer
- by James Keesling PDF
- Proc. Amer. Math. Soc. 41 (1973), 315-320 Request permission
Abstract:
Let $C$ denote the category of compact Hausdorff spaces and continuous maps and $H:C \to HC$ the homotopy functor to the homotopy category. Let $S:C \to SC$ denote the functor of shape in the sense of Holsztyński for the projection functor $H$. Every continuous mapping $f$ between spaces gives rise to a shape morphism $S(f)$ in $SC$, but not every shape morphism is in the image of $S$. In this paper it is shown that if $X$ is a continuum with $x \in X$ and $A$ is a compact connected abelian topological group, then if $F$ is a shape morphism from $X$ to $A$, then there is a continuous map $f:X \to A$ such that $f(x) = 0$ and $S(f) = F$. It is also shown that if $f,g:X \to A$ are continuous with $f(x) = g(x) = 0$ and $S(f) = S(g)$, then $f$ and $g$ are homotopic. These results are then used to show that there are shape classes of continua containing no locally connected continua and no arcwise connected continua. Some other applications to shape theory are given also.References
- K. Borsuk, On a locally non-movable continuum, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 425–430 (English, with Russian summary). MR 256356
- A. M. Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950), 35–43. MR 33830, DOI 10.1090/S0002-9939-1950-0033830-7 K. H. Hofmann, Introduction to the theory of compact groups. I, Tulane University Lecture Notes, Tulane University, New Orleans, La., 1968.
- Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
- W. Holsztyński, Continuity of Borsuk’s shape functor, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1105–1108 (English, with Russian summary). MR 310873
- 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
- James Keesling, Shape theory and compact connected abelian topological groups, Trans. Amer. Math. Soc. 194 (1974), 349–358. MR 345064, DOI 10.1090/S0002-9947-1974-0345064-8
- 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. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 0201557
- 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
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 315-320
- MSC: Primary 54C56
- DOI: https://doi.org/10.1090/S0002-9939-1973-0334141-8
- MathSciNet review: 0334141