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The number of smooth 4-manifolds with a fixed complexity

Dave Auckly
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Abstract

One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as fast as $n^{c\sqrt[3]{n}}$. Along the way we construct complete kirby diagrams for a large family of knot surgery manifolds.

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Dave Auckly. 2007-06-18. The number of smooth 4-manifolds with a fixed complexity. https://arxiv.org/abs/math/0701269

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