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Tori in the diffeomorphism groups of simply-connected 4-manifolds

Published online by Cambridge University Press:  24 October 2008

Paul Melvin
Paul Melvin
Affiliation:
University of California, Santa Barbara
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Extract

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff(M) isomorphic to the 2-dimensional torus T2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T2-actions on M. Thus the classification problem is trivial unless M supports an effective T2-action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ;P2 and S2 × S2 (2), and so this paper is really a study of the different T2-actions on these manifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 2 , March 1982 , pp. 305 - 314
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Melvin, P.On 4-manifolds with singular torus actions. Math. Ann. 256 (1981) 255–276.CrossRefGoogle Scholar
(2)Orlik, P. and Raymond, F.Actions of the torus on 4-manifolds I. Trans. Amer. Math. Soc. 152 (1970) 531–559Google Scholar