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Transverse regularity for maps of homology manifolds

Published online by Cambridge University Press:  24 October 2008

N. Martin
N. Martin
Affiliation:
Downing College, Cambridge
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Extract

Recall that in (2) we showed that it was possible to make transverse a homology manifold and PL-manifold inside a large dimensional homology manifold, subject to being able to do some general position inside the large manifold. In (1) we were able to relax the condition that one of the submanifolds be a PL-manifold to it being a homotopy manifold. The way in which the making transverse was achieved was via a system of h-cobordisms from the original situation to the transverse one. The problem we tackle here is that of making a map between homology manifolds transverse regular. Thus we ask: given a map f: M → N of homology manifolds with P a proper submanifold of N, is it possible to homotop f to a map g: M → N such that g−1(P) is a proper submanifold of M and g induces a map from the normal bundle of g−1(P) in M to the normal bundle of P in N?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 29 - 38
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Martin, N. Polyhedral homology manifolds: Thesis, Cambridge, 1971.Google Scholar
(2)Martin, N.Cobordism of homology manifolds. Proc. Cambridge Philos. Soc. 71 (1972), 247–270.CrossRefGoogle Scholar
(3)Martin, N.On the difference between homology and piecewise-linear bundles. J. London Math. Soc. (to appear).Google Scholar
(4)Martin, N. and Maunder, C. R. F.Homology cobordism bundles: Topology, 10 (1971), 92–110.CrossRefGoogle Scholar
(5)Maunder, C. R. F.On the Pontryagin classes of homology manifolds. Topology, 10 (1971), 111–118.CrossRefGoogle Scholar
(6)Maunder, C. R. F.Algebraic Topology (Van Nostrand, London, 1970).Google Scholar
(7)Maunder, C. R. F.General position theorems for homology manifolds. J. London Math. Soc. 4 (1972), 760–768.CrossRefGoogle Scholar
(8)Maunder, C. R. F.Improving the general position properties of homology manifolds. Proc. London Math. Soc. 25 (1972), 689–700.CrossRefGoogle Scholar
(9)Rourke, C. P. and Sanderson, B. J.Δ-sets II. Quart. J. Math. Oxford Ser. 22 (1971), 465–486.CrossRefGoogle Scholar
(10)Spivax, M.Spaces satisfying Poincare duality. Topology 6 (1967), 77–104.CrossRefGoogle Scholar
(11)Stallings, J. R.Lectures on polyhedral topology. Tata Institute of Fundamental Research (Bombay, 1968).Google Scholar
(12)Wall, C. T. C.Surgery on compact manifolds (Academic Press; London, 1970).Google Scholar