Absolutely exotic compact 4-manifolds

We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds h...

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Veröffentlicht in:Commentarii mathematici Helvetici 2016-01, Vol.91 (1), p.1-19
Hauptverfasser: Akbulut, Selman, Ruberman, Daniel
Format: Artikel
Sprache:eng
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Zusammenfassung:We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, absolute means that the exotic structure is not relative to a particular parameterization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible homology cobordism along its boundary. Applying this technique to corks (contractible manifolds with a diffeomorphism of the boundary that does not extend to a diffeomorphism of the interior) gives examples of absolutely exotic smooth structures on contractible 4-manifolds.
ISSN:0010-2571
1420-8946
DOI:10.4171/CMH/375