Geometry and Topology

Manifolds: Varieties, and Knots

Martin A. Mccrory

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000153934
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This book discusses topics ranging from traditional areas of topology, such as knot theory and the topology of manifolds, to areas such as differential and algebraic geometry. It also discusses other topics such as three-manifolds, group actions, and algebraic varieties.

Automorphisms of Punctured-Surface Bundles

DARRYL MCCULLOUGH / Department of Mathematics, University of Oklahoma, Norman, Oklahoma

0. INTRODUCTION

Let S be a connected orientable surface of genus g ≥ 1 with one boundary component, and let ϕ be a self-homeomorphism of S. The mapping torus M(ϕ) = S × I/((x,0) ∼ (ϕ(x),1)) is a 3-manifold which fibers over S¹ with S as fiber. Such manifolds arise naturally as complements of fibered knots in closed orientable 3-manifolds. In fact, any closed orientable 3-manifold contains a fibered knot [22], [14] whose complement M(ϕ) has a hyperbolic structure [13].

When g = 1, M(ϕ) is a punctured-torus bundle. In the orientable case these manifolds have been studied in [12], [11], and [24]; in particular the incompressible surfaces in M(ϕ) can be completely understood. In [12] it is observed that the mapping class groups Γ(M(ϕ)) can be computed. In the first three sections of the present work, we carry this out in both the orientable and nonorientable cases. The results are summarized in section 4.

In section 5, still in the punctured-torus case, we analyze the effects of mapping classes on the boundary of M(ϕ). The fundamental group of ∂M(ϕ) is generated by the boundary of a fiber, called c, and a section to the fibering on ∂M(ϕ), called t. While c is determined up to orientation, a choice of t, called a framing of M(ϕ) in [11], must be made. All horneomorphisms must carry c to c^±1 (since c generates the kernel of H₁(∂M) → H₁(M)) and hence must carry t to t^±1c_k for some k ∈ Z. In section 5 we come to a complete understanding of the restriction homomorphism Γ(M(ϕ)) → Γ(∂M(ϕ)).

When a solid torus is attached to (an orientable) M(ϕ) so that the meridian of the solid torus is attached along t, the result is a closed orientable 3-manifold N(ϕ), and the core of the solid torus is a fibered knot K in N(ϕ) whose genus equals the genus of S. When M(ϕ) has a homeomorphism carrying t to t^±1c_k , it is clear that ±l/k surgery on N(ϕ) along k yields a manifold homeomorphic to N(ϕ). In particular, if N(ϕ) = S³ and such a homeomorphism exists, then K would be a nontrivial knot in S³ which does not have property P [3]. Since it is known [10] that this could only happen if k = ±1, we focus on this case. Motivated by the analysis of the punctured-torus case in section 5, we give in section 6 constructions of many M(ϕ) having homeomorphisms carrying t to (tc)^±1 , so that +1 surgery along K yields N(ϕ). For g = 3 we find some examples where N(ϕ) is a Z-homology 3-sphere. However, we also prove that our constructions can never yield N(ϕ) = S³.

The reader whose main interest is the surgery examples will have no difficulty starting in section 6 after a cursory reading...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Preface
• Table of Contents
• Contributors
• Introduction to resolution towers
• A geometric construction of the boundedly controlled Whitehead group
• An introduction to boundedly controlled simple homotopy theory
• On certain branched cyclic covers of S3
• A geometric interpretation of Siebenmann’s periodicity phenomenon
• Regular convex cell complexes
• Gauge theory and smooth structures on 4-manifolds
• Algebraic varieties which are a disjoint union of subvarieties
• The lower central series of generalized pure braid groups
• Implication of the geometrization conjecture for the algebraic K-theory of 3-manifolds
• On the diffeomorphism types of certain elliptic surfaces
• Deformations of flat bundles over Kähler manifolds
• Imbeddings of knot groups in knot groups
• Geometric Hopfian and non-Hopfian situations
• Deformations of totally geodesic foliations
• Automorphisms of punctured-surface bundles
• Lattice gauge fields and Chern-Weil theory
• Intrinsic skeleta and intersection homology of weakly stratified sets
• Isolated critical points of maps from to R4 to R2 and a natural splitting of the Milnor number of a classical fibred link, part II
• Covering theorems for open surfaces
• Equivariant handles in finite group actions
• The role of knot theory in DNA research
• Continuous versus discrete symmetry
• The knotting of theta curves and other graphs in S3
• Index