Abstract
In this paper, we describe the history and the present status of one of the main classical problems in low-dimensional geometric topology—the recognition of topological 3-manifolds in the class of all generalized 3-manifolds (i.e., ANR homology 3-manifolds). This problem naturally splits into the cell-like resolution problem for 3-manifolds by means of homology 3-manifolds and the general-position problem for topological 3-manifolds. We have also included some open problems.
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References
R. H. Bing, “A decomposition of E 3 into points and tame arcs such that the decomposition space is topologically different from E 3,” Ann. Math. (2), 65, 484–500 (1957).
M. V. Brahm, “Approximating maps of 2-manifolds with zero-dimensional nondegeneracy sets,” Topology Appl., 45, 25–38 (1992).
M. G. Brin, Doctoral dissertation, Univ. of Wisconsin, Madison (1977).
M. G. Brin, “Generalized 3-manifolds whose non-manifold set has neighborhoods bounded by tori,” Trans. Amer. Math. Soc., 264, 539–555 (1981).
M. G. Brin and D. R. McMillan, Jr., “Generalized three-manifolds with zero-dimensional non-manifold set,” Pacific J. Math., 97, 29–58 (1981).
E. M. Brown, M. S. Brown, and C. D. Feustel, “On properly embedding planes in 3-manifolds,” Proc. Amer. Math. Soc., 55, 461–464 (1976).
J. Bryant, S. Ferry, W. Mio, and S. Weinberger, “Topology of homology manifolds,” Ann. Math. (2), 143, 435–467 (1996).
J. L. Bryant and R. C. Lacher, “Resolving acyclic images of three-manifolds,” Math. Proc. Cambridge Philos. Soc., 88, 311–319 (1980).
J. W. Cannon, “The recognition problem: What is a topological manifold?” Bull. Amer. Math. Soc., 84, 832–866 (1978).
A. Cavicchioli, “Imbeddings of polyhedra in 3-manifolds,” Ann. Mat. Pura Appl., 162, 157–177 (1992).
A. Cavicchioli, F. Hegenbarth, and D. Repovš, “On the construction of 4k-dimensional generalized manifolds,” in: F. T. Farrell and W. Luck, eds., Proc. School ICTP “ High-Dimensional Manifold Topology” (Trieste, Italy, 21 May-8 June 2001 ), Vol. 2 (2003), pp. 103–124.
A. Cavicchioli, F. Hegenbarth, and D. Repovš, Topology of Cell-like Maps and Homology Manifolds (in preparation).
A. Cavicchioli and D. Repovš, “Peripheral acyclicity and homology manifolds,” Ann. Mat. Pura Appl., 172, 5–24 (1997).
R. J. Daverman and D. Repovš, “A new 3-dimensional shrinking theorem,” Trans. Amer. Math. Soc., 315, 219–230 (1989).
R. J. Daverman and D. Repovš, “General position properties which characterize 3-manifolds, ” Can. J. Math., 44, 234–251 (1992).
R. J. Daverman and W. H. Row, “Cell-like 0-dimensional decompositions of S 3 are 4-manifold factors,” Trans. Amer. Math. Soc., 254, 217–236 (1979).
R. J. Daverman and T. L. Thickstun, “The 3-manifold recognition problem,” Trans. Amer. Math. Soc., to appear.
R. D. Edwards, Approximating Certain Cell-Like Maps by Homeomorphisms, manuscript, UCLA, Los Angeles (1977).
J. Hempel, 3-Manifolds, Princeton Univ. Press, Princeton (1986).
H. W. Lambert and R. B. Sher, “Pointlike 0-dimensional decompositions of S 3,” Pacific J. Math., 24, 511–518 (1968).
V. A. Nicholson, “1-FLG complexes are tame in 3-manifolds,” General Topology Appl., 2, 277–285 (1972).
F. S. Quinn, “An obstruction to the resolution of homology manifolds,” Michigan Math. J., 34, 285–291 (1987).
D. Repovš, “Recognition problem for manifolds,” in: J. Krasinkiewicz, S. Spiez, and H. Toruńczyk, eds., Geometric and Algebraic Topology, Banach Centre Publ., Vol. 18, PWN, Warsaw (1986), pp. 77–108.
D. Repovš and R. C. Lacher, “A disjoint disks property for 3-manifolds,” Topology Appl., 16, 161–170 (1983).
T. L. Thickstun, “An extension of the loop theorem and resolutions of generalized 3-manifolds with 0-dimensional singular set,” Invent. Math., 78, 161–222 (1984).
T. L. Thickstun, “Strongly acyclic maps and homology 3-manifolds with 0-dimensional singular set,” Proc. London Math. Soc. (3), 55, 378–432 (1987).
T. L. Thickstun, “Resolutions of generalized 3-manifolds whose singular sets have general position dimension one,” Topology Appl., 138, 61–95 (2004).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 71–84, 2005.
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Cavicchioli, A., Repovš, D. & Thickstun, T.L. Geometric topology of generalized 3-manifolds. J Math Sci 144, 4413–4422 (2007). https://doi.org/10.1007/s10958-007-0279-y
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DOI: https://doi.org/10.1007/s10958-007-0279-y