Elliptic Three-Manifolds and the Smale Conjecture

Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

After a discussion of the (Generalized) Smale Conjecture, the main results of the monograph are summarized. The extent to which the Smale Conjecture extends to larger classes of three-manifolds—usually in a limited form called the Weak Smale Conjecture, if at all—is detailed. The chapter closes with a brief discussion of why Perelman’s methods appear not to give progress on the Smale Conjecture. As noted in the Preface, theSmale Conjecture is the assertion that the inclusion is a homotopy equivalence whenever M is an elliptic three-manifold, that is, a three-manifold with a Riemannian metric of constant positive curvature (which may be assumed to be 1). TheGeometrization Conjecture, now proven byPerelman, shows that all closed three-manifolds with finite fundamental group are elliptic.In this chapter, we will first review elliptic three-manifolds and their isometry groups. In the second section, we will state our main results on the Smale Conjecture, and provide some historical context. In the final two sections, we discuss isometries of nonelliptic three-manifolds, and address the possibility of applying Perelman’s methods to the Smale Conjecture.

Original languageEnglish
Title of host publicationDiffeomorphisms of Elliptic 3-Manifolds
PublisherSpringer Verlag
Pages1-7
Number of pages7
ISBN (Print)9783642315633
DOIs
Publication statusPublished - 2012 Jan 1

Publication series

NameLecture Notes in Mathematics
Volume2055
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

Keywords

  • Fundamental Group
  • Isometry Group
  • Klein Bottle
  • Lens Space
  • Mapping Class Group

ASJC Scopus subject areas

  • Algebra and Number Theory

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  • Cite this

    Hong, S., Kalliongis, J., McCullough, D., & Rubinstein, J. H. (2012). Elliptic Three-Manifolds and the Smale Conjecture. In Diffeomorphisms of Elliptic 3-Manifolds (pp. 1-7). (Lecture Notes in Mathematics; Vol. 2055). Springer Verlag. https://doi.org/10.1007/978-3-642-31564-0_1