A new 3-dimensional shrinking criterion

Daverman, Robert J.; Repovš, Dušan

doi:10.1090/S0002-9947-1989-0978381-X

A new $3$-dimensional shrinking criterion
HTML articles powered by AMS MathViewer

by Robert J. Daverman and Dušan Repovš PDF

Trans. Amer. Math. Soc. 315 (1989), 219-230 Request permission

Abstract:

We introduce a new shrinking criterion for cell-like upper semicontinuous decompositions $G$ of topological $3$-manifolds, such that the embedding dimension (in the sense of Štan’ko) of the nondegeneracy set of $G$ is at most one. As an immediate application, we prove a recognition theorem for $3$-manifolds based on a new disjoint disks property.

References

E. H. Anderson, Approximation of certain continuous functions of $S^{2}$ into $E^{3}$, Proc. Amer. Math. Soc. 18 (1967), 889–891. MR 216471, DOI 10.1090/S0002-9939-1967-0216471-0
Steve Armentrout, Homotopy properties of decomposition spaces, Trans. Amer. Math. Soc. 143 (1969), 499–507. MR 273574, DOI 10.1090/S0002-9947-1969-0273574-9
Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
J. L. Bryant and R. C. Lacher, Resolving acyclic images of three-manifolds, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 311–319. MR 578275, DOI 10.1017/S0305004100057613
J. W. Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), no. 5, 832–866. MR 494113, DOI 10.1090/S0002-9904-1978-14527-3
Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
R. J. Daverman and W. H. Row, Cell-like $0$-dimensional decompositions of $S^{3}$ are $4$-manifold factors, Trans. Amer. Math. Soc. 254 (1979), 217–236. MR 539916, DOI 10.1090/S0002-9947-1979-0539916-8
Robert D. Edwards, Demension theory. I, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 195–211. MR 0394678

Approximating certain cell-like maps by homeomorphisms

George Kozlowski and John J. Walsh, Cell-like mappings on $3$-manifolds, Topology 22 (1983), no. 2, 147–151. MR 683754, DOI 10.1016/0040-9383(83)90025-3
R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), no. 4, 495–552. MR 645403, DOI 10.1090/S0002-9904-1977-14321-8
R. C. Lacher, Generalized three-manifolds, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 82–92. MR 643524
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
D. R. McMillan Jr., Compact, acyclic subsets of three-manifolds, Michigan Math. J. 16 (1969), 129–136. MR 243501
Frank Quinn, Ends of maps. III. Dimensions $4$ and $5$, J. Differential Geometry 17 (1982), no. 3, 503–521. MR 679069
Dušan Repovš, The recognition problem for topological manifolds, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 77–108. MR 925857

dimensional cell-like decompositions of

manifolds

Dušan Repovš and R. C. Lacher, A disjoint disks property for $3$-manifolds, Topology Appl. 16 (1983), no. 2, 161–170. MR 712862, DOI 10.1016/0166-8641(83)90016-0
Dušan Repovš and R. C. Lacher, Resolving acyclic images of nonorientable three-manifolds, Proc. Amer. Math. Soc. 90 (1984), no. 1, 157–161. MR 722436, DOI 10.1090/S0002-9939-1984-0722436-2
L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. MR 295365, DOI 10.1016/0040-9383(72)90014-6
Michael Starbird, Cell-like $0$-dimensional decompositions of $E^{3}$, Trans. Amer. Math. Soc. 249 (1979), no. 1, 203–215. MR 526318, DOI 10.1090/S0002-9947-1979-0526318-3
M. A. Štan′ko, Imbedding of compacta in euclidean space, Mat. Sb. (N.S.) 83 (125) (1970), 234–255 (Russian). MR 0271923
John J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 105–118. MR 643526