Constructing infinitely many smooth structures on small 4-manifolds

Abstract

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ for $k=1,...,4$, or to $3\CP# (2l+3)\CPb$ for $l=1,...,6$. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on $\CP#3\CPb$, $3\CP#5\CPb$ and $3\CP#7\CPb$.

23 pages, 3 figures

References in corpus (3)

• Simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$
• An exotic smooth structure on CP^2+6CP^2-bar
• Reverse engineering small 4-manifolds

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