Abstract:A compact subset $X$ of a polyhedron $P$ is cellular in $P$ if there is a pseudoisotopy of $P$ shrinking precisely $X$ to a point. A proper surjection $f:P\rightarrow Q$ is cellular if each point inverse of $f$ is cellular in $P$. 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.
- Ethan Akin, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413–473. MR 253329, DOI 10.1090/S0002-9947-1969-0253329-1 F. D. Ancel, Engulfing the tracks of a proper homotopy, notes.
- Steve Armentrout, Cellular decompositions of $3$-manifolds that yield $3$-manifolds, Memoirs of the American Mathematical Society, No. 107, American Mathematical Society, Providence, R.I., 1971. MR 0413104
- J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), no. 1, 83–112. MR 541330, DOI 10.2307/1971245 R. D. Edwards, Approximating certain cellular maps by homeomorphisms, manuscript.
- Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
- Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), no. 1, 287–338 (1980). MR 583711
- Michael Handel, Approximating stratum preserving CE maps between CS sets by stratum preserving homeomorphisms, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 245–250. MR 0407854
- James P. Henderson, Cellularity in polyhedra, Topology Appl. 12 (1981), no. 3, 267–282. MR 623735, DOI 10.1016/0166-8641(81)90005-5
- James P. Henderson, Approximating cellular maps between low-dimensional polyhedra, Pacific J. Math. 101 (1982), no. 2, 321–331. MR 675404
- James P. Henderson, A discussion of results and problems related to cellularity in polyhedra, Continua, decompositions, manifolds (Austin, Tex., 1980) Univ. Texas Press, Austin, Tex., 1983, pp. 141–151. MR 711986
- C. Lacher, Locally flat strings and half-strings, Proc. Amer. Math. Soc. 18 (1967), 299–304. MR 212805, DOI 10.1090/S0002-9939-1967-0212805-1
- D. R. McMillan Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327–337. MR 161320, DOI 10.2307/1970548
- L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. MR 295365, DOI 10.1016/0040-9383(72)90014-6 —, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 74 (1972), 123-163.
- John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488. MR 149457
- © Copyright 1982 American Mathematical Society
- 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