Mathematical Research Letters

Volume 2 (1995)

Number 4

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Pages: 497 – 503

DOI: http://dx.doi.org/10.4310/MRL.1995.v2.n4.a9

Author

Richard Stong (Rice University)

Abstract

We will prove that any closed, simply-connected smooth 4-manifold admits a handlebody structure where the 2-handles homotopically cancel the 1-handles and the dual 1-handles in the nicest possible way. As a consequence we will derive the following improved splitting theorem for closed, simply-connected smooth 4-manifolds. Suppose $M$\ is a closed, simply-connected, smooth 4-manifold and the intersection form of $M$\ splits as $(H_2(M;Z), q_M ) \cong (Z^{n_1}, \lambda_1 )\dsum(Z^{n_2}, \lambda_2 )$. Then there is a decomposition $M\cong M_1\cup_\Sigma M_2$, where $M_i$ is a compact, simply-connected, smooth 4-manifold with boundary the homology 3-sphere $\Sigma$ and intersection form isomorphic to $(Z^{n_i}, \lambda_i )$. This result is also extended to 4-manifolds with free fundamental groups.

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