⭕Geometric Group Theory Unit 11 – Applications to 3–Manifolds

Key Concepts in 3-Manifolds

• 3-manifolds are topological spaces locally homeomorphic to Euclidean 3-space ($\mathbb{R}^3$)
• Compact 3-manifolds can be classified into several categories based on their geometric and topological properties
  • Examples include spherical 3-manifolds ($S^3$), hyperbolic 3-manifolds, and Seifert-fibered 3-manifolds
• Every closed, orientable 3-manifold can be obtained by performing Dehn surgery on a link in the 3-sphere
• The fundamental group of a 3-manifold encodes essential information about its topology and geometry
• Geometric structures on 3-manifolds, such as hyperbolic, spherical, or Euclidean structures, provide additional insights into their properties
• The study of 3-manifolds has important applications in low-dimensional topology, knot theory, and other areas of mathematics

Fundamental Groups and 3-Manifolds

• The fundamental group of a 3-manifold $M$, denoted $\pi_1(M)$, is a powerful invariant that captures the essential topological features of the manifold
• For a closed, orientable 3-manifold, the fundamental group is always finitely presentable
• The fundamental group can be used to distinguish between different 3-manifolds and to study their covering spaces
  • For example, the 3-sphere has a trivial fundamental group, while the 3-torus has a fundamental group isomorphic to $\mathbb{Z}^3$
• The geometrization conjecture, proven by Perelman, states that every closed, orientable 3-manifold can be decomposed into pieces, each admitting one of eight geometric structures
• The fundamental group of a 3-manifold is related to its geometric structure
  • Hyperbolic 3-manifolds have fundamental groups that are hyperbolic, while Seifert-fibered 3-manifolds have fundamental groups that are extensions of surface groups by $\mathbb{Z}$
• The study of fundamental groups of 3-manifolds has led to important developments in geometric group theory and low-dimensional topology

Geometric Structures on 3-Manifolds

• A geometric structure on a 3-manifold is a complete, locally homogeneous Riemannian metric
• The eight Thurston geometries that can occur on closed, orientable 3-manifolds are $S^3$, $E^3$, $H^3$, $S^2 \times \mathbb{R}$, $H^2 \times \mathbb{R}$, $\widetilde{SL_2(\mathbb{R})}$, Nil, and Sol
  • Each geometry corresponds to a unique simply connected Riemannian manifold with a specific isometry group
• Hyperbolic 3-manifolds admit a complete Riemannian metric of constant negative curvature
• Seifert-fibered 3-manifolds are foliated by circles and admit one of the six remaining Thurston geometries
• The existence of a geometric structure on a 3-manifold has significant implications for its topology and geometry
  • For instance, hyperbolic 3-manifolds are always aspherical and have infinite fundamental groups
• The study of geometric structures on 3-manifolds is closely related to the geometrization conjecture and has applications in various areas of mathematics

Thurston's Geometrization Conjecture

• Thurston's geometrization conjecture, formulated by William Thurston in the 1970s, provides a comprehensive framework for understanding the geometry and topology of 3-manifolds
• The conjecture states that every closed, orientable 3-manifold can be decomposed into pieces, each admitting one of eight geometric structures
  • The eight geometries are $S^3$, $E^3$, $H^3$, $S^2 \times \mathbb{R}$, $H^2 \times \mathbb{R}$, $\widetilde{SL_2(\mathbb{R})}$, Nil, and Sol
• The conjecture was proven by Grigori Perelman in 2003 using the Ricci flow technique
• The geometrization conjecture generalizes the Poincaré conjecture, which states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere
• The proof of the geometrization conjecture has far-reaching consequences in 3-manifold topology and geometric group theory
  • It provides a complete classification of closed, orientable 3-manifolds and has led to significant advances in the understanding of their properties
• The geometrization conjecture has inspired further research in related areas, such as the study of 4-manifolds and the development of new geometric and topological techniques

Hyperbolic 3-Manifolds

• Hyperbolic 3-manifolds are 3-manifolds that admit a complete Riemannian metric of constant negative curvature
• They are one of the most important classes of 3-manifolds and play a central role in the geometrization conjecture
• Hyperbolic 3-manifolds are always aspherical, meaning their universal cover is contractible
• The fundamental groups of hyperbolic 3-manifolds are hyperbolic groups, which have rich algebraic and geometric properties
  • Hyperbolic groups are finitely presentable, have a linear isoperimetric inequality, and satisfy the Tits alternative
• The volume of a hyperbolic 3-manifold is a topological invariant and can be used to distinguish between different manifolds
• The study of hyperbolic 3-manifolds has connections to various areas of mathematics, including number theory, complex analysis, and representation theory
  • For example, the volumes of hyperbolic 3-manifolds are related to special values of L-functions and the Chern-Simons invariant

Knot Theory and 3-Manifolds

• Knot theory is the study of embeddings of circles in 3-dimensional space, up to ambient isotopy
• Every knot complement (the 3-manifold obtained by removing a tubular neighborhood of the knot from $S^3$) is a 3-manifold with boundary
• The fundamental group of a knot complement, known as the knot group, encodes important information about the knot
  • The knot group can be used to distinguish between different knots and to study their properties, such as chirality and genus
• Dehn surgery on knots is a powerful tool for constructing 3-manifolds
  • Every closed, orientable 3-manifold can be obtained by performing Dehn surgery on a link in the 3-sphere
• The geometrization conjecture has significant implications for knot theory
  • It implies that every knot complement can be decomposed into pieces, each admitting one of the eight Thurston geometries
• The study of knots and their complements has led to important developments in 3-manifold topology and geometric group theory

Applications to Low-Dimensional Topology

• The study of 3-manifolds has numerous applications in low-dimensional topology, which is the study of manifolds of dimension 4 or less
• The geometrization conjecture provides a complete classification of closed, orientable 3-manifolds, which is a major milestone in the field
• The techniques developed in the study of 3-manifolds, such as Dehn surgery and the use of geometric structures, have been applied to the study of 4-manifolds
  • For example, the construction of exotic smooth structures on 4-manifolds often involves the use of 3-manifold techniques
• The study of knots and their complements is a central topic in low-dimensional topology
  • Knot invariants, such as the Jones polynomial and the Alexander polynomial, have important applications in the classification of 3-manifolds and the study of their properties
• The fundamental groups of 3-manifolds and their representations have been used to study the topology of low-dimensional manifolds
  • For instance, the virtual Haken conjecture, which states that every closed, irreducible 3-manifold has a finite-sheeted cover that is Haken, has been proven using techniques from geometric group theory
• The study of 3-manifolds and their applications in low-dimensional topology continues to be an active area of research, with many open questions and conjectures driving further developments in the field

Connections to Other Areas of Mathematics

• The study of 3-manifolds has deep connections to various other areas of mathematics, including algebra, geometry, analysis, and mathematical physics
• Hyperbolic 3-manifolds have been used to construct examples of expander graphs, which have applications in computer science and network theory
• The volumes of hyperbolic 3-manifolds are related to special values of L-functions, linking the study of 3-manifolds to number theory and arithmetic geometry
• The Chern-Simons invariant, which is defined using the geometry of 3-manifolds, has important applications in mathematical physics, particularly in the study of topological quantum field theories
• The study of geometric structures on 3-manifolds has led to the development of new techniques in geometric analysis, such as the Ricci flow and the study of Einstein metrics
• The fundamental groups of 3-manifolds and their representations have connections to the theory of von Neumann algebras and the study of operator algebras
  • For example, the Kazhdan property (T) for groups, which has important applications in representation theory and ergodic theory, has been studied in the context of 3-manifold groups
• The study of 3-manifolds has also inspired the development of new algebraic and topological techniques, such as the use of quantum invariants and the study of categorified knot invariants
• As the field of 3-manifold topology continues to evolve, it is likely that new connections to other areas of mathematics will emerge, further highlighting the central role that 3-manifolds play in modern mathematics