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A characterization of piecewise linear homology manifolds

Published online by Cambridge University Press:  24 October 2008

W. J. R. Mitchell
W. J. R. Mitchell
Affiliation:
Magdalene College, Cambridge
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Extract

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 2 , March 1983 , pp. 271 - 274
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Bing, R. H.A characterization of three-space by partitionings. Trans. Amer. Math. Soc. 70 (1951), 15–27.CrossRefGoogle Scholar
(2)Bryant, J. L. and Lacher, R. C.Resolving zero-dimensional singularities in generalized manifolds. Math. Proc. Cambridge Philos. Soc. 83 (1978), 403–413.CrossRefGoogle Scholar
(3)Cannon, J. W.The recognition problem: what is a topological manifold? Bull. Amer. Math. Soc. 84 (1978), 832–866.CrossRefGoogle Scholar
(4)Edwards, R. D. The topology of manifolds and cell-like maps. Proc. International. Cong. Mathematicians (Helsinki 1978), vol. 1, 111–127.Google Scholar
(5)Fary, I.Absolute subcomplexes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1977), 433–471.Google Scholar
(6)Galewski, D. E. and Stern, R. J.Classification of simplicial triangulations of topological manifolds. Ann. of Maths. 111 (1980), 1–34.CrossRefGoogle Scholar
(7)van Kampen, E. R.On some characterisations of two-dimensional manifolds. Duke Math. Jour. 1 (1935), 74–93.CrossRefGoogle Scholar
(8)Kervaire, M.Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 (1969), 67–72.CrossRefGoogle Scholar
(9)Kirby, R. C. and Siebenmann, L. C.Foundational essays on topological manifolds, smoothings and triangulations. Ann. of Math. Stud. 88 (Princeton, 1977).Google Scholar
(10)Martin, N.Cobordism of homology manifolds. Proc. Cambridge Philos. Soc. 71 (1972), 247–270.CrossRefGoogle Scholar
(11)Martin, N.On the difference between homology and piecewise linear bundles. J. London Math. Soc. (2) 6 (1973), 197–204.CrossRefGoogle Scholar
(12)Mitchell, W. J. R. General position properties of ANR's. (To appear in Math. Proc. Cambridge Philos. Soc.)Google Scholar
(13)Quinn, F.Ends of maps 1. Ann. of Maths. 110 (1979), 275–331.CrossRefGoogle Scholar
(14)Quinn, F. Resolutions of homology manifolds (An informal sketch). Polycopied notes.Google Scholar
(15)Siebenmann, L. C. The obstruction to finding a boundary for an open manifold of dimension ≥ 5. Thesis (Princeton 1965).Google Scholar
(16)Siebenmann, L. C.On detecting open collars. Trans. Amer. Math. Soc. 142 (1969), 201–227.CrossRefGoogle Scholar
(17)Stallings, J. R.Lectures on polyhedral topology. (Tata Institute, Bombay 1968).Google Scholar