Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T08:46:21.166Z Has data issue: false hasContentIssue false
  • English
  • Français

An exotic factor of S3 × ℝ

Published online by Cambridge University Press:  24 October 2008

W. Jakobsche  and
D. Repovš
W. Jakobsche
Affiliation:
University of Warsaw, 00-901 Warsaw, Poland
D. Repovš
Affiliation:
University of Ljubljana, 61000 Ljubljana, Yugoslavia
Get access Rights & Permissions [Opens in a new window]

Extract

Cannon's recognition problem [10] asks for a short list of topological properties that is reasonably easy to check and that characterizes topological manifolds. In dimensions below three the answer has been known for a long time: see [6, 24]. In dimensions above four it is now known, due to the work of J. W. Cannon [11], R. D. Edwards [14] (see also [12] and [18]), and F. S. Quinn [21], that topological n-manifolds (n ≥ 5) are precisely ENR ℤ-homology n-manifolds with Cannon's disjoint disc property (DDP) [11] and with a vanishing Quinn's local surgery obstruction [23]. In dimension four there is a resolution theorem of Quinn [22] (with the same obstruction as in dimensions ≥ 5) and a 1-LCC shrinking theorem of M. Bestvina and J. J. Walsh [5]. However, it is still an open problem to find an effective analogue of Cannon's DDP for this dimension, one which would yield a shrinking theorem along the lines of that of Edwards [14]. For more on the history of the recognition problem see the survey [24].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 329 - 344
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ancel, F. D.. An alternative proof of M. Brown's theorem on inverse sequences of near homeomorphisms. In Geometric Topology and Shape Theory (eds. Mardešić, S. and Segal, J.), Lecture Notes in Math. vol. 1283 (Springer-Verlag, 1987), pp. 1–2.CrossRefGoogle Scholar
[2]Ancel, F. D. and Siebenmann, L. C.. The construction of homogeneous homology manifolds. Abstracts. Amer. Math. Soc. 6 (1985), 92.Google Scholar
[3]Armentrout, S.. Cellular Decompositions of 3-manifolds that yield 3-manifolds. Memoirs Amer. Math. Soc. no. 107 (American Mathematical Society, 1971).CrossRefGoogle Scholar
[4]Bessaga, C. and Pelczyński, A.. Infinite-dimensional Topology. Monogr. Math. no. 58 (Polish Scientific Publishers, 1975).Google Scholar
[5]Bestvina, M. and Walsh, J. J.. A 1-LCC shrinking theorem in dimension 4. (Preprint, UCLA, Los Angeles, 1988.)Google Scholar
[6]Bing, R. H.. The Kline sphere characterization problem. Bull. Amer. Math. Soc. 52 (1946), 644–653.CrossRefGoogle Scholar
[7]Bing, R. H.. Approximating surfaces with polyhedral ones. Ann. of Math. (2) 65 (1957), 456–483.CrossRefGoogle Scholar
[8]Brown, M.. Some applications of an approximation theorem for inverse limits. Proc. Amer. Math. Soc. 11 (1960), 478–483.CrossRefGoogle Scholar
[9]Bryant, J. L. and Lacher, R. C.. Resolving acyclic images of 3-manifolds. Math. Proc. Cambridge Philos. Soc. 88 (1980), 311–320.CrossRefGoogle Scholar
[10]Cannon, J. W., The recognition problem: What is a topological manifold? Bull. Amer. Math. Soc. 84 (1978), 832–866.CrossRefGoogle Scholar
[11]Cannon, J. W.. Shrinking cell-like decompositions of manifolds: Codimension three. Ann. of Math. (2) 110 (1979), 83–112.CrossRefGoogle Scholar
[12]Daverman, R. J.. Decompositions of Manifolds (Academic Press, 1986).Google Scholar
[13]Davis, M. W.. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2) 117 (1983). 293–324.CrossRefGoogle Scholar
[14]Edwards, R. D.. The topology of manifolds and cell-like maps. In Proceedings International Congress of Mathematicians, Helsinki 1978 (Acad. Sci. Fennica, 1980), pp. 111–127.Google Scholar
[15]Jakobsche, W.. The Bing-Borsuk Conjecture is stronger than the Poincaré Conjecture. Fund. Math. 106 (1980), 127–134.CrossRefGoogle Scholar
[15a]Jakobsche, W.. Homogeneous cohomology manifolds which ate inverse limits. (Preprint, University of Varsaw, 1989.)Google Scholar
[16]Lacher, R. C.. Cell-like mappings and their generalizations. Bull. Amer. Math. Soc. 83 (1977), 495–552.CrossRefGoogle Scholar
[17]Lamber, H. V. and Sher, R. B.. Pointlike 0-dimensional decompositions of S 3. Pacific J. Math. 24 (1968), 511–518.CrossRefGoogle Scholar
[18]Latour, F.. Double suspension d'une sphère d'homologie (d'après R. Edwards). In Séminar Bourbaki 1977/78, Lecture Notes in Math. vol. 710 (Springer-Verlag, 1979), pp. 169–186.Google Scholar
[19]Mitchell, W. J. R., Repovš, D. and Ščepin, E. V.. A geometric criterion for the finite dimensionality of cell-like quotients of 4-manifolds. (Preprint, University of Cambridge, 1988.)Google Scholar
[20]Pontryagin, L. S.. Sur une hypothèse fondamentale de la théorie de la dimension Comptes Rend. Acad. Sci. Paris 190 (130), 1105–1107.Google Scholar
[21]Quinn, F. S.. Resolutions of homology manifolds, and the topological characterization of manifolds. Invent. Math. 72 (1983), 267–284. Erratum 85 (1986). 653.CrossRefGoogle Scholar
[22]Quinn, F. S.. Ends of maps, III: dimensions 4 and 5 (includes the annulus conjecture). J. Diff. Geom. 17 (1982), 503–521.Google Scholar
[23]Quinn, F. S.. An obstruction to the resolution of homology manifolds. Michigan Math. J. 34 (1987), 285–291.CrossRefGoogle Scholar
[24]Repovš, D.. The recognition problem for topological manifolds. In Algebraic and Geometric Topology, Banach Center Publ. no. 18 (Polish Scientific Publishers, 1986). pp. 77–108.Google Scholar
[25]Repovš, D. and Lacher, R. C.. A disjoint disks property for 3-manifolds. Topol. Appl. 16 (1983), 161–170.CrossRefGoogle Scholar
[26]Siebenmann, L. C.. Approximating cellular maps by homeomorphisms. Topology 11 (1972), 271–294.CrossRefGoogle Scholar
[27]Walsh, J. J.. The finite dimensionality of integral homology 3-manifolds. Proc. Amer. Math. Soc. 88 (1983), 154–156.CrossRefGoogle Scholar
[28]Williams, R. F.. A useful functor and three famous examples in topology. Trans. Amer. Math. Soc. 106 (1963). 319–329.CrossRefGoogle Scholar