#### Volume 20, issue 7 (2020)

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A note on the complexity of $h$–cobordisms

### Hannah R Schwartz

Algebraic & Geometric Topology 20 (2020) 3313–3327
##### Abstract

We show that the number of double points of smoothly immersed $2$–spheres representing certain homology classes of an oriented, smooth, closed, simply connected $4$–manifold $X$ must increase with the complexity of corresponding $h$–cobordisms from $X$ to $X\phantom{\rule{-0.17em}{0ex}}$. As an application, we give results restricting the minimal number of double points of immersed spheres in manifolds homeomorphic to rational surfaces.

##### Keywords
topology, manifold, smooth structures, spheres, $h$–cobordism
##### Mathematical Subject Classification 2010
Primary: 57Q20, 57Q99
##### Publication
Received: 1 December 2018
Revised: 19 January 2020
Accepted: 2 April 2020
Published: 29 December 2020
##### Authors
 Hannah R Schwartz Max Planck Institute of Mathematics Bonn Germany