13.1 Statement and proof of the h-cobordism theorem

The h-cobordism theorem is a powerful result in differential topology. It connects smooth structures on manifolds to algebraic properties of homotopy equivalences, providing a way to classify certain types of manifolds up to diffeomorphism.

This theorem uses tools from Morse theory, like handle decompositions and critical point cancellation. It also involves topological concepts such as simple connectivity and Whitehead torsion, bridging different areas of mathematics in a surprising way.

Cobordisms and Diffeomorphisms

Cobordisms and their Properties

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• h-cobordism: A cobordism $(W; M_0, M_1)$ where the inclusions $M_0 \hookrightarrow W$ and $M_1 \hookrightarrow W$ are homotopy equivalences
• Cobordism: A compact manifold $W$ whose boundary is the disjoint union of two closed manifolds $M_0$ and $M_1$, denoted as $(W; M_0, M_1)$
  • Provides a smooth transition between two manifolds
  • Can be thought of as a "bridge" connecting $M_0$ and $M_1$
• Homotopy equivalence: A continuous map between two topological spaces that has a homotopy inverse
  • If $M_0$ and $M_1$ are homotopy equivalent to $W$, they are homotopy equivalent to each other
  • Homotopy equivalence preserves important topological properties (homology, homotopy groups)

Diffeomorphisms and Smooth Structures

• Diffeomorphism: A smooth bijective map between manifolds whose inverse is also smooth
  • Diffeomorphic manifolds have the same smooth structure
  • h-cobordisms can be used to study the existence of diffeomorphisms between manifolds
  • Example: The torus and the square with opposite sides identified are diffeomorphic

Morse Theory Tools

Morse Functions and Critical Points

• Morse function: A smooth real-valued function on a manifold whose critical points are non-degenerate
  • Critical points correspond to topological changes in the manifold
  • Morse functions can be used to study the topology of a manifold
  • Example: The height function on a torus has four critical points (maximum, minimum, and two saddles)
• Gradient-like vector field: A vector field that is compatible with a Morse function
  • Integral curves of a gradient-like vector field connect critical points
  • Used to define the flow on a manifold associated with a Morse function

Handle Decompositions and Cancellation

• Handle decomposition: A way to build a manifold by attaching handles of various indices to a disk
  • Each handle corresponds to a critical point of a Morse function
  • The index of a handle is the number of "negative" directions of the Hessian at the corresponding critical point
  • Example: A 2-dimensional handle decomposition of a torus consists of one 0-handle, two 1-handles, and one 2-handle
• Cancellation of critical points: A process of eliminating pairs of critical points in a Morse function
  • Cancellation is possible when the indices of the critical points differ by 1
  • Cancellation simplifies the handle decomposition and the topology of the manifold

Topological Conditions

Simple Connectivity and the Fundamental Group

• Simply connected: A topological space is simply connected if it is path-connected and has a trivial fundamental group
  • The fundamental group measures the number of distinct loops in a space that cannot be continuously deformed into each other
  • Simply connected spaces have no non-trivial loops
  • Example: The 2-sphere is simply connected, while the torus is not

Whitehead Torsion and h-Cobordisms

• Whitehead torsion: An algebraic invariant associated with a homotopy equivalence between CW complexes
  • Measures the "twisting" of the homotopy equivalence
  • Vanishing of Whitehead torsion is a necessary condition for a homotopy equivalence to be a simple homotopy equivalence
  • In the h-cobordism theorem, the vanishing of Whitehead torsion ensures that the cobordism is trivial (a product cobordism)
  • Example: The Whitehead torsion of the identity map on a CW complex is zero