Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T17:12:34.778Z Has data issue: false hasContentIssue false
  • English
  • Français

General position properties of ANR's

Published online by Cambridge University Press:  24 October 2008

W. J. R. Mitchell
W. J. R. Mitchell
Affiliation:
Magdalene College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 451 - 466
Copyright
Copyright © Cambridge Philosophical Society 1982

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)Bing, R. H. and Borsuk, K.Locally homogeneous spaces. Annals of Maths. 81 (1965), 100–111.CrossRefGoogle Scholar
(2)Borsuk, K.Theory of retracts. Monog. Mat. 44 (Warsaw, 1967).Google Scholar
(3)Bredon, G. E.Sheaf theory (McGraw-Hill, 1967).Google Scholar
(4)Bredon, G. E. Generalised manifolds, revisited. Proceedings of the Georgia Conference, 1969, ed. Cantrell, J. C. and Edwards, C. H. (Markham, 1970), pp. 461–469.Google Scholar
(5)Cannon, J. W.Shrinking cell-like decompositions of manifolds; codimension three. Annals of Maths. 110 (1979), 83–112.CrossRefGoogle Scholar
(6)Cannon, J. W. and Daverman, R. J.A totally wild flow. Indiana Univ. J. Math. 30 (1981), 371–387.CrossRefGoogle Scholar
(7)Chapman, T. A.Lectures on Hilbert Cube Manifolds. Amer. Math. Soc. Regional Conferences Series, no. 28 (1976).CrossRefGoogle Scholar
(8)Daverman, R. J.Detecting the disjoint discs property. Pacific J. Math. 93 (1981), 277–299.CrossRefGoogle Scholar
(9)Daverman, R. J. and Walsh, J. J.Čech homology characterisations of infinite dimensional manifolds Amer. J. Math. 103 (1981), 411–436.CrossRefGoogle Scholar
(10)De Groot, J.On the topological characterisation of manifolds. Third Prague Topology Symposium, 1971, pp. 155–158.Google Scholar
(11)Edwards, R. D.Demension Theory, I. Proceedings of Park City conf., ed. Glaser, L. C. and Rushing, T. B., Lecture Notes in Maths, vol. 438 (Springer, 1975), pp. 193–211.Google Scholar
(12)Edwards, R. D. The topology of manifolds and cell-like maps. (Proceedings of the Inter-national Congress of Mathematicians, Helsinki 1978, vol. 1, 111–126).Google Scholar
(13)Hu, S-T.Retracts (Wayne State U. Press, 1965).Google Scholar
(14)Hurewicz, W. and Wallman, H.Dimension theory (Princeton, 1948).Google Scholar
(15)Jakobsche, W.The Bing-Borsuk conjecture is stronger than the Poincaré conjecture. Fund Math. 106 (1980), 127–134.CrossRefGoogle Scholar
(16)Lay, T. L. and Walsh, J. J.ANR's Admitting an Interval Factor are Q-manifolds. Proc. Amer. Math. Soc. 73 (1979), 279–280.Google Scholar
(17)Mitchell, W. J. R.Absolute suspensions and cones. Fund. Math. 101 (1978), 241–244.CrossRefGoogle Scholar
(18)Quinn, F.Ends of maps. I. Annals of Maths. 110 (1979), 275–331.CrossRefGoogle Scholar
(19)Quinn, F. Resolutions of homology manifolds, an informal sketch. (Preprint.)Google Scholar
(20)Szymański, A.Remarks on the absolute suspension. Fund. Math. 86 (1974), 157–161.CrossRefGoogle Scholar
(21)Singh, S.A linking axiom and retracts. Bull. de l'Acad. Polon. des Sciences (Math. Astron. Phys.) 26 (1978), 817–823.Google Scholar
(22)Toruńczyk, H.On CE-images of the Hilbert cube and characterisation of Q-manifolds. Fundamenta Math. 106 (1980), 31–40.CrossRefGoogle Scholar
(23)Ungar, S.On local homology. Glasnik Mat. 14 (34) (1979), 151–158.Google Scholar
(24)Wilder, R. L.Monotone mappings of manifolds. Pacific J. Math. 7 (1957), 1519–1528.CrossRefGoogle Scholar
(25)Willard, S.General topology (Addison-Wesley 1970).Google Scholar