Cellular maps between polyhedra

Author:
James P. Henderson

Journal:
Trans. Amer. Math. Soc. **272** (1982), 527-537

MSC:
Primary 57Q55; Secondary 54E60, 57N60, 57Q37

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662050-3

MathSciNet review:
662050

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Abstract | References | Similar Articles | Additional Information

Abstract: A compact subset of a polyhedron is cellular in if there is a pseudoisotopy of shrinking precisely to a point. A proper surjection is cellular if each point inverse of is cellular in . We give certain conditions under which cellular maps between polyhedra are approximable by homeomorphisms. An example of a cellular map which is not approximable by homeomorphisms is also given.

**[1]**Ethan Aiken,*Manifold phenomena in the theory of polyhedra*, Trans. Amer. Math. Soc.**143**(1969), 413-473. MR**0253329 (40:6544)****[2]**F. D. Ancel,*Engulfing the tracks of a proper homotopy*, notes.**[3]**S. Armentrout,*Cellular decompositions of**-manifolds that yield**-manifolds*, Mem. Amer. Math. Soc. No.**107**(1971). MR**0413104 (54:1225)****[4]**J. W. Cannon,*Shrinking cell-like decompositions of manifolds: codimension three*, Ann. of Math. (2)**110**(1979), 83-112. MR**541330 (80j:57013)****[5]**R. D. Edwards,*Approximating certain cellular maps by homeomorphisms*, manuscript.**[6]**R. D. Edwards and R. C. Kirby,*Deformations of spaces of imbeddings*, Ann. of Math. (2)**93**(1971), 63-88. MR**0283802 (44:1032)****[7]**R. Geoghegan,*Open problems in infinite-dimensional topology*, Topology Proc.**4**(1979), 287-338. MR**583711 (82a:57015)****[8]**Michael Handel,*Approximating stratum preserving CE maps between CS sets of stratum preserving homeomorphisms*, Geometric Topology (L. C. Glaser and T. B. Rushing, Eds.), Lecture Notes in Math., vol. 438, Springer-Verlag, New York, 1975, pp. 245-250. MR**0407854 (53:11624)****[9]**J. Henderson,*Cellularity in polyhedra*, Topology Appl.**12**(1981), 267-282. MR**623735 (83b:57014)****[10]**-,*Approximating cellular maps between low dimensional polyhedra*, Pacific J. Math. (to appear). MR**675404 (84b:57012)****[11]**-,*A discussion of results and problems related to cellularity in polyhedra*, Proc. 1980 Summer Topology Conference (Austin, Texas) (to appear). MR**711986 (84i:57016)****[12]**R. C. Lacher,*Locally flat strings and half strings*, Proc. Amer. Math. Soc.**18**(1967), 299-304. MR**0212805 (35:3670)****[13]**D. R. McMillan, Jr.,*A criterion for cellularity in a manifold*, Ann. of Math. (2)**79**(1964), 327-337. MR**0161320 (28:4528)****[14]**L. C. Siebenmann,*Approximating cellular maps by homeomorphisms*, Topology**11**(1972), 271-294. MR**0295365 (45:4431)****[15]**-,*Deformations of homeomorphisms on stratified sets*, Comment. Math. Helv.**74**(1972), 123-163.**[16]**J. R. Stallings,*The piecewise-linear structure of Euclidean space*, Proc. Cambridge Philos. Soc.**58**(1962), 481-488. MR**0149457 (26:6945)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662050-3

Keywords:
Cellular map,
polyhedron,
approximation,
stratification

Article copyright:
© Copyright 1982
American Mathematical Society