Property $SUV^{\infty }$ and proper shape theory
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- by R. B. Sher PDF
- Trans. Amer. Math. Soc. 190 (1974), 345-356 Request permission
Abstract:
A class of spaces called the $SU{V^\infty }$ spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the $U{V^\infty }$ spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an $SU{V^\infty }$ space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the $SU{V^\infty }$ spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property $U{V^n}$, called $SU{V^n}$, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an $SU{V^n}$ space if and only if X is an $SU{V^\infty }$ space. Finally, it is shown that a locally finite connected simplicial complex is an $SU{V^\infty }$ space if and only if all of its homotopy and proper homotopy groups vanish.References
- Steve Armentrout, $\textrm {UV}$ properties of compact sets, Trans. Amer. Math. Soc. 143 (1969), 487β498. MR 273573, DOI 10.1090/S0002-9947-1969-0273573-7
- B. J. Ball and R. B. Sher, A theory of proper shape for locally compact metric spaces, Bull. Amer. Math. Soc. 79 (1973), 1023β1026. MR 334139, DOI 10.1090/S0002-9904-1973-13312-9
- B. J. Ball and R. B. Sher, A theory of proper shape for locally compact metric spaces, Fund. Math. 86 (1974), 163β192. MR 365472, DOI 10.4064/fm-86-2-163-192
- Karol Borsuk, Fundamental retracts and extensions of fundamental sequences, Fund. Math. 64 (1969), 55-85; errata, ibid. 64 (1969), 375. MR 0243520, DOI 10.4064/fm-64-1-55-85
- Edward M. Brown, Proper homotopy theory in simplicial complexes, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp.Β 41β46. MR 0356041
- Hans Freudenthal, Neuaufbau der Endentheorie, Ann. of Math. (2) 43 (1942), 261β279 (German). MR 6504, DOI 10.2307/1968869 Dean Hartley, Quasi-cellularity in manifolds, Ph.D. thesis, University of Georgia, Athens, Ga., 1973. (Condensed version, to appear).
- J. R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR 0170323, DOI 10.1090/surv/012
- W. Kuperberg, Homotopically labile points of locally compact metric spaces, Fund. Math. 73 (1971/72), no.Β 2, 133β136. MR 298641, DOI 10.4064/fm-73-2-133-136
- R. C. Lacher, Cell-like spaces, Proc. Amer. Math. Soc. 20 (1969), 598β602. MR 234437, DOI 10.1090/S0002-9939-1969-0234437-3
- D. R. McMillan Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327β337. MR 161320, DOI 10.2307/1970548 β, UV properties and related topics, mimeographed lecture notes, Florida State University, Tallahassee, Fla., 1970.
- Sibe MardeΕ‘iΔ, Retracts in shape theory, Glasnik Mat. Ser. III 6(26) (1971), 153β163 (English, with Serbo-Croatian summary). MR 296915
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 190 (1974), 345-356
- MSC: Primary 54C56
- DOI: https://doi.org/10.1090/S0002-9947-1974-0341389-0
- MathSciNet review: 0341389