The Geometry of Spherical Space Form Groups

Peter B Gilkey

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813220805
• Edition: 2
• Topic: Mathematics
• Subtopic: Vector Analysis
• Index: Mathematics

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This volume focuses on discussing the interplay between the analysis, as exemplified by the eta invariant and other spectral invariants, the number theory, as exemplified by the relevant Dedekind sums and Rademacher reciprocity, the algebraic topology, as exemplified by the equivariant bordism groups, K-theory groups, and connective K-theory groups, and the geometry of spherical space forms, as exemplified by the Smith homomorphism. These are used to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group.

This volume is a completely rewritten revision of the first edition. The underlying organization is modified to provide a better organized and more coherent treatment of the material involved. In addition, approximately 100 pages have been added to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group. We have chosen to focus on the geometric aspect of the theory rather than more abstract algebraic constructions (like the assembly map) and to restrict our attention to spherical space forms rather than more general and more complicated geometrical examples to avoid losing contact with the fundamental geometry which is involved.

--> Contents:

• Partial Differential Operators
• K Theory and Cohomology
• Equivariant Bordism
• Positive Scalar Curvature
• Auxiliary Materials

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--> Readership: Graduate students and researchers interested in global analysis, geometry, and topology. -->
Keywords:Dedekind Sums and Rademacher Reciprocity;K-Theory;Eta Invariant;Spherical Space Form;Lens Space;Quaternion Spherical Space Form;Iterated Jet Bundle;Equivariant Bordism;Smith Homomorphism;Connective K-Theory;Manifolds with Positive Scalar Curvature;Spin Bordism;Unitary Bordism;Spin-C Bordism;Pin-C BordismReview: Key Features:

• The is a complete revision of the first edition and includes substantial amounts of new material applying the basic material of the book to the examination of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group
• To ensure that the book is accessible to wide an audience as possible, there is a review of vector bundle theory, of Clifford module theory, of the Atiyah–Singer index theorem, and of the index theorem with boundary
• There are also tables, which have been simplified and the organization improved from the first edition, giving various K-theory and equivariant bordism groups