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A Note on Doubles of 4-Manifolds

Published online by Cambridge University Press:  20 November 2018

Steven H. Weintraub
Steven H. Weintraub*
Affiliation:
Department of Mathematics Louisiana State University and Agricultural & Mechanical College Baton Rouge, Louisiana 70803
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If M is a simply-connected 4-manifold with boundary, let D(M) denote its double MU∂M(-M). If M is closed, let D(M) denote M#-M. In either case, D(M) is a simply-connected 4-manifold of index zero, and so by a theorem of Wall [8], M#k(S2xS2) must be standard for k sufficiently large, where by standard we mean diffeomorphic to the connected sum of copies of S2 x S2 and S2×S2, the non-trivial S2 bundle over S2 (which is itself diffeomorphic to ℂP2#-ℂP2 [7]). In this paper we give abound on k, in the case where M has no 3-handles.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 3 , 01 September 1980 , pp. 367 - 369
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Andrews, J.J. and Curtis, M.L., Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192-195.Google Scholar
2. Harer, J., Kas, A., and Kirby, R., to appear.Google Scholar
3. Kaplan, S., Constructing framed 4-manifolds with given almost framed boundaries, Dissertation, Berkeley, 1976.Google Scholar
4. Kirby, R., ed., Problems in Low-Dimensional Manifold Theory, Proc. Symp. in Pure Math, xxxii, v. 2, 273-312, A.M.S. Providence, R.I., 1978.Google Scholar
5. Lickorish, W. B. R., A Representation of Orientable Combinatorial 3-manifolds, Ann. of Math. 76 (1962), 531-540.Google Scholar
6. Mandelbaum, R., Special Handlebody Decompositions of Simply-Connected Algebraic Surfaces, to appear.Google Scholar
7. Neumann, W.D. and Weintraub, S.H., Four-manifolds constructed via plumbing, Math. Ann. 238 (1978)71-78.Google Scholar
8. Wall, C. T. C., On simply-connected four-manifolds, J. London Math. Soc, 39 (1964), 141-149.Google Scholar