Decompositions into codimension-two manifolds

Daverman, R. J.; Walsh, J. J.

doi:10.1090/S0002-9947-1985-0773061-4

Decompositions into codimension-two manifolds
HTML articles powered by AMS MathViewer

by R. J. Daverman and J. J. Walsh PDF

Trans. Amer. Math. Soc. 288 (1985), 273-291 Request permission

Abstract:

Let $M$ denote an orientable $(n + 2)$-manifold and let $G$ denote an upper semicontinuous decomposition of $M$ into continua having the shape of closed, orientable $n$-manifolds. The main result establishes that the decomposition space $M/G$ is a $2$-manifold.

References

Edward G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. (2) 51 (1950), 534–543. MR 35015, DOI 10.2307/1969366
D. S. Coram and P. F. Duvall Jr., Mappings from $S^{3}$ to $S^{2}$ whose point inverses have the shape of a circle, General Topology Appl. 10 (1979), no. 3, 239–246. MR 546098
D. S. Coram and P. F. Duvall Jr., Finiteness theorems for approximate fibrations, Trans. Amer. Math. Soc. 269 (1982), no. 2, 383–394. MR 637696, DOI 10.1090/S0002-9947-1982-0637696-9
R. J. Daverman, Decompositions of manifolds into codimension one submanifolds, Compositio Math. 55 (1985), no. 2, 185–207. MR 795714
R. J. Daverman and L. S. Husch, Decompositions and approximate fibrations, Michigan Math. J. 31 (1984), no. 2, 197–214. MR 752256, DOI 10.1307/mmj/1029003024
R. J. Daverman and J. J. Walsh, Decompositions into codimension two spheres and approximate fibrations, Topology Appl. 19 (1985), no. 2, 103–121. MR 789592, DOI 10.1016/0166-8641(85)90064-1

Decompositions into submanifolds that yield generalized manifolds

Local

connectivity of decomposition spaces

D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI 10.1112/plms/s3-16.1.369
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493

Manifolds accepting codimension one sphere-like decompositions

Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Andrzej Szulkin, $\textbf {R}^{3}$ is the union of disjoint circles, Amer. Math. Monthly 90 (1983), no. 9, 640–641. MR 719756, DOI 10.2307/2323284
G. T. Whyburn, Interior Transformations on Surfaces, Amer. J. Math. 60 (1938), no. 2, 477–490. MR 1507329, DOI 10.2307/2371311
Raymond Louis Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications, Vol. XXXII, American Mathematical Society, Providence, R.I., 1963. MR 0182958