🗺️Morse Theory Unit 8 – Handlebodies and Handle Decompositions

Definition and Basics

• A handlebody is a 3-dimensional manifold with boundary obtained by attaching solid handles to a 3-ball
• The genus of a handlebody is the number of handles attached to the 3-ball
  • A handlebody of genus $g$ is homeomorphic to the regular neighborhood of a bouquet of $g$ circles in $\mathbb{R}^3$
• The boundary of a handlebody is a closed orientable surface of genus $g$
• Handlebodies are compact, connected, and orientable 3-manifolds
• The core of a handle is a circle that goes through the center of the handle, connecting the attaching regions on the boundary of the 3-ball
• The co-core of a handle is a disk that intersects the core transversely at a single point

Handlebody Construction

• Start with a 3-ball $B^3$ as the base space for constructing a handlebody
• Attach 1-handles (solid cylinders) to the boundary of the 3-ball by identifying the ends of the cylinder with disjoint disks on the boundary
  • The attaching regions for 1-handles are disjoint disks on the boundary of the 3-ball
• The attaching process involves removing the interior of the attaching disks and identifying the ends of the 1-handle with the resulting boundary components
• Multiple 1-handles can be attached to the 3-ball to create handlebodies of higher genus
• The order in which the 1-handles are attached does not affect the resulting handlebody up to homeomorphism
• Attaching $g$ 1-handles to a 3-ball results in a handlebody of genus $g$

Handle Attachments

• Handles are attached to a manifold by identifying the boundary of a handle with a specific region on the boundary of the manifold
• The dimension of the handle determines the type of attachment and the resulting change in the manifold's topology
  • 0-handles are attached by identifying the boundary of a 3-ball with a single point on the manifold
  • 1-handles are attached by identifying the boundary of a solid cylinder (1-handle) with two disjoint disks on the manifold's boundary
  • 2-handles are attached by identifying the boundary of a 2-handle (a 2-disk bundle over a 1-disk) with a circle on the manifold's boundary
  • 3-handles are attached by identifying the boundary of a 3-ball with a 2-sphere on the manifold's boundary
• The attaching region for a handle is a submanifold of the boundary of the manifold being attached to
• The core of a handle is a submanifold of the handle that captures its essential topology (a point for 0-handles, a circle for 1-handles, a disk for 2-handles, and a 2-sphere for 3-handles)
• The attaching process involves removing the interior of the attaching region and identifying the boundary of the handle with the resulting boundary components

Types of Handles

• 0-handles are 3-balls attached to a single point on the manifold
  • Attaching a 0-handle creates a new connected component in the manifold
• 1-handles are solid cylinders attached to two disjoint disks on the manifold's boundary
  • Attaching a 1-handle increases the genus of the manifold by 1 or connects two separate boundary components
• 2-handles are 2-disk bundles over 1-disks attached along a circle on the manifold's boundary
  • Attaching a 2-handle decreases the genus of the manifold by 1 or separates a single boundary component into two
• 3-handles are 3-balls attached to a 2-sphere on the manifold's boundary
  • Attaching a 3-handle fills in a spherical boundary component and decreases the number of connected components by 1
• The dimension of a handle refers to the dimension of its core submanifold (0 for 0-handles, 1 for 1-handles, etc.)

Handle Decomposition Process

• A handle decomposition of a manifold is a way to build the manifold by attaching handles of increasing dimension
• Start with a collection of 0-handles (disjoint 3-balls)
• Attach 1-handles to the 0-handles by identifying pairs of disks on their boundaries
  • This process creates a handlebody with genus equal to the number of 1-handles attached
• Attach 2-handles to the resulting handlebody along circles on its boundary
  • Attaching 2-handles can decrease the genus or split the boundary into multiple components
• Attach 3-handles to fill in any remaining spherical boundary components
• The order in which handles of the same dimension are attached does not affect the resulting manifold up to homeomorphism
• A handle decomposition provides a systematic way to understand the topology of a manifold by breaking it down into simpler pieces

Morse Functions on Handlebodies

• A Morse function on a handlebody is a smooth real-valued function with non-degenerate critical points
• Critical points of a Morse function correspond to handles in a handle decomposition of the handlebody
  • Index 0 critical points correspond to 0-handles
  • Index 1 critical points correspond to 1-handles
  • Index 2 critical points correspond to 2-handles
  • Index 3 critical points correspond to 3-handles
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
• Morse functions can be used to construct handle decompositions of handlebodies
  • As the value of the Morse function increases, handles are attached to the sublevel sets at critical points
• The attaching regions for handles can be determined by the flow lines of the gradient vector field of the Morse function
• Morse functions provide a way to relate the topology of a handlebody to the critical points of a smooth function

Applications in Topology

• Handlebodies and handle decompositions are fundamental tools in the study of 3-manifolds and 4-manifolds
• Every compact, orientable 3-manifold admits a handle decomposition
  • This allows for the classification of 3-manifolds using techniques from Morse theory and handle decompositions
• Handle decompositions can be used to compute topological invariants such as homology and cohomology groups
  • The chain complex associated with a handle decomposition provides a way to calculate these invariants
• Kirby diagrams, which represent handle decompositions of 4-manifolds, are used to study and classify 4-manifolds
• The Morse inequalities relate the number of critical points of a Morse function to the Betti numbers of the manifold
  • This provides a connection between the topology of a manifold and the critical points of smooth functions on it
• Handle sliding and cancellation techniques are used to simplify handle decompositions and study the relationships between different decompositions of the same manifold

Examples and Exercises

• Construct a handlebody of genus 2 by attaching two 1-handles to a 3-ball
  • Describe the attaching regions and the resulting boundary surface
• Given a Morse function on a handlebody, determine the corresponding handle decomposition
  • Identify the critical points and their indices, and describe the attaching process for each handle
• Compute the homology groups of a handlebody using its handle decomposition
  • Set up the chain complex associated with the decomposition and calculate the homology groups
• Describe the handle decomposition of a solid torus and its boundary surface
  • Identify the types of handles involved and their attaching regions
• Create a handle decomposition of the 3-sphere by attaching a single 0-handle and a single 3-handle
  • Explain why this decomposition is valid and how it relates to the topology of the 3-sphere
• Construct a genus 2 handlebody using a Morse function with two index 1 critical points
  • Describe the level sets of the Morse function and how they relate to the attached handles