Connections, Definite Forms, and Four-Manifolds

ISBN-13: 9780198535997 / Angielski / Twarda / 1991 / 142 str.

Ted Petrie; John Randall

Opis

The central theme of this book is the study of self-dual connections on four-manifolds. The authors have adopted a topologists' perspective and so have included many of the explicit calculations using the Atiyah-Singer index theorem as well as presenting arguments couched in terms of equivariant topology. The authors' aim is to present moduli space techniques applied to four-manifolds and to study vector bundles over four-manifolds whose structure group is SO(3).Results covered include Donaldson's proof that the only positive definite forms occur as intersection forms and the results of Fintushel and Stern which show that the integral homology cobordism group of integral homology three-spheres has elements of infinite order. Little previous knowledge of differential geometry is assumed and so postgraduate students and research workers will find this both an accessible and complete introduction to an area that is currently one of the most active in mathematical research.

The central theme of this book is the study of the relationship between the geometry and topology of four-manifolds. The authors' aim is to present a lucid introduction to moduli space techniques (for vector bundles with SO(3) as structure group) and to apply them to four-manifolds. The authors have adopted a topologists' perspective. They have included explicit calculations using the Atiyah-Singer index theorem; results covered include Donaldson's Theorem and those of Fintushel and Stern.

Recenzje

'For topologists it might be the easiest way into parts of the theory which was started by Donaldson.' P. Michor, Moatshefte für Mathematik, Vol. 112, 1991, No. 3

Spis treści

Preface; Introduction; Connections; SO(3) - connections; Index of the fundamental complex; The virtual moduli space B; The virtual moduli space M; Intersection forms on 4-manifolds; Moduli space for invariant connections; Applications to homology 3-spheres; Appendices; Bibliography.