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Decompositions into codimension-two manifolds. (English) Zbl 0568.57013
This paper establishes the result that the decomposition space of a usc decomposition of an orientable \((n+2)\)-manifold into continua having the shape of closed orientable n-manifolds is itself a 2-manifold.
From the introduction: ”This theorem lifts what is known about decompositions into codimension-two submanifolds to nearly the same level as what is known about decompositions into codimension-one submanifolds.” In particular, this paper extends work by Liem and by Daverman on codimension-one decompositions and a previous joint paper by the same authors [Topology Appl. 19, 103-121 (1985)]. They state that the present work evolved from techniques of their previous paper, ”which were inspired for the most part by methods of Coram and Duvall”.
Finally, the authors point out that the relatively nice results for decompositions into codimension-one or -two submanifolds ”do not persist through successively larger codimensions”. For codimension three, the decomposition space need not even be a generalized manifold.
Reviewer: L.Cannon

MSC:
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N25 Shapes (aspects of topological manifolds)
54B15 Quotient spaces, decompositions in general topology
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[1] Edward G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. (2) 51 (1950), 534 – 543. · Zbl 0036.38803 · doi:10.2307/1969366 · doi.org
[2] D. S. Coram and P. F. Duvall Jr., Mappings from \?³ to \?² whose point inverses have the shape of a circle, General Topology Appl. 10 (1979), no. 3, 239 – 246. · Zbl 0417.54014
[3] D. S. Coram and P. F. Duvall Jr., Finiteness theorems for approximate fibrations, Trans. Amer. Math. Soc. 269 (1982), no. 2, 383 – 394. · Zbl 0489.55013
[4] R. J. Daverman, Decompositions of manifolds into codimension one submanifolds, Compositio Math. 55 (1985), no. 2, 185 – 207. · Zbl 0593.57005
[5] R. J. Daverman and L. S. Husch, Decompositions and approximate fibrations, Michigan Math. J. 31 (1984), no. 2, 197 – 214. · Zbl 0584.57011 · doi:10.1307/mmj/1029003024 · doi.org
[6] R. J. Daverman and J. J. Walsh, Decompositions into codimension two spheres and approximate fibrations, Topology Appl. 19 (1985), no. 2, 103 – 121. · Zbl 0589.57012 · doi:10.1016/0166-8641(85)90064-1 · doi.org
[7] -, Decompositions into submanifolds that yield generalized manifolds (in preparation). · Zbl 0617.57009
[8] J. Dydak and J. Segal, Local \( n\)-connectivity of decomposition spaces (to appear). · Zbl 0547.54027
[9] D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369 – 383. · Zbl 0148.43103 · doi:10.1112/plms/s3-16.1.369 · doi.org
[10] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119 – 221 (French). · Zbl 0118.26104
[11] Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. · Zbl 0060.39808
[12] V.-T. Liem, Manifolds accepting codimension one sphere-like decompositions (to appear). · Zbl 0582.57009
[13] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. · Zbl 0145.43303
[14] Andrzej Szulkin, \?³ is the union of disjoint circles, Amer. Math. Monthly 90 (1983), no. 9, 640 – 641. · Zbl 0521.52011 · doi:10.2307/2323284 · doi.org
[15] G. T. Whyburn, Interior Transformations on Surfaces, Amer. J. Math. 60 (1938), no. 2, 477 – 490. · JFM 64.0612.05 · doi:10.2307/2371311 · doi.org
[16] Raymond Louis Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications, Vol. XXXII, American Mathematical Society, Providence, R.I., 1963. · Zbl 0039.39602
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