Torelli Actions and Smooth Structures on 4-manifolds

Abstract

Artin presentations are discrete equivalents of planar open book decompositions of closed, orientable three manifolds. Artin presentations characterize the fundamental groups of closed, orientable three manifolds. An Artin presentation also determines a smooth, compact, simply conected four manifold that bounds the three dimensional open book. In this way, the study of three and four manifolds may be approached purely group theoretically. In the theory of Artin presentations, elements of the Torelli subgroup act on the topology and smooth structures of the three and four manifolds. We show that the Torelli action can preserve the continuous topological type of a four manifold while changing its smooth structure. This is a new, group theoretic method of altering the smooth structure on a four manifold.

Publisher

World Scientific Publishing

Publication Date

1-1-2008

Publication Title

Journal of Knot Theory and Its Ramifications

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1142/S0218216508006075

Language

English

Format

text

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