🗺️Morse Theory Unit 13 – The h–Cobordism Theorem

Key Concepts and Definitions

• Morse function $f: M \to \mathbb{R}$ smooth function on a smooth manifold $M$ whose critical points are non-degenerate
• Critical point $p \in M$ point where the differential $df_p = 0$
  • Non-degenerate critical point has non-singular Hessian matrix
• Index of a critical point number of negative eigenvalues of the Hessian matrix at that point
• Cobordism $(W; M_0, M_1)$ compact manifold $W$ with boundary $\partial W = M_0 \sqcup M_1$
  • $h$-cobordism has $M_0, M_1$ homotopy equivalent to $W$
• Morse lemma local form of a Morse function near a non-degenerate critical point
• Gradient vector field $\nabla f$ assigns each point the direction of steepest ascent of $f$
• Stable and unstable manifolds submanifolds consisting of points that flow to/from a critical point under $-\nabla f$

Historical Context and Development

• Marston Morse (1892-1977) American mathematician, pioneer of Morse theory
• Morse theory originated in 1920s as a tool in calculus of variations
• Key early result Morse inequalities relating critical points to homology (1930s)
• Raoul Bott (1923-2005) made significant contributions, including Bott periodicity (1950s)
• Stephen Smale proved the h-cobordism theorem using Morse theory (1960)
  • Fundamental result in high-dimensional topology
• Morse theory has since found wide-ranging applications in different areas of mathematics
  • Floer homology, Witten's approach to Morse theory, applications in physics

Fundamental Principles of Morse Theory

• Study of smooth manifolds using critical points of smooth functions
• Non-degenerate critical points isolated, classified by their index
• Morse functions admit nice local coordinates (Morse lemma)
  • Quadratic form $\pm x_1^2 \pm \cdots \pm x_n^2$
• Gradient vector field $\nabla f$ encodes dynamics of the function
• Stable/unstable manifolds of a critical point diffeomorphic to $\mathbb{R}^k, \mathbb{R}^{n-k}$
  • $k$ index of the critical point, $n$ dimension of the manifold
• Changes in topology occur only at critical points
• Morse inequalities relate number of critical points to Betti numbers
  • $m_k \geq b_k$, $m_k$ number of critical points of index $k$

The h-Cobordism Theorem Statement

• Let $(W; M_0, M_1)$ be a compact $h$-cobordism of dimension $n \geq 6$
  • $W, M_0, M_1$ simply connected
• Then $W$ is diffeomorphic to $M_0 \times [0,1]$
  • Cobordism is trivial, $M_0$ and $M_1$ are diffeomorphic
• Equivalently, if $M_0, M_1$ are homotopy equivalent, high-dimensional, simply connected closed manifolds
  • Then any $h$-cobordism between them is trivial
• Theorem does not hold in dimensions $n \leq 3$
  • Counterexamples exist in dimension 4 (E8 manifold) and 5

Proof Outline and Key Steps

1. Choose Morse function $f: W \to [0,1]$ with $f^{-1}(0) = M_0, f^{-1}(1) = M_1$
   • Rearrange critical values to be distinct in $(0,1)$
2. Analyze changes in level sets $W_t = f^{-1}([0,t])$ as $t$ increases
   • Crossing a critical point of index $k$ attaches a $k$-handle
3. Use homotopy equivalences to show all critical points have index 0 or $n$
   • 0-handles and $n$-handles correspond to $M_0 \times D^n$ and $D^n \times M_1$
4. Introduce a partial order on the set of critical points
   • Smale's cancellation theorem allows canceling pairs of handles
5. Rearrange and cancel handles until only one 0-handle and one $n$-handle remain
   • Requires dimension $n \geq 6$ for sufficient room to maneuver
6. Resulting cobordism is trivial, diffeomorphic to $M_0 \times [0,1]$

Applications and Implications

• Provides classification of simply connected, high-dimensional manifolds
  • Up to h-cobordism, determined by homotopy type
• Implies generalized Poincaré conjecture in dimensions $n \geq 6$
  • Every homotopy $n$-sphere is homeomorphic to the $n$-sphere $S^n$
• Used in the proof of the topological invariance of Whitehead torsion
• Fundamental tool in surgery theory and classification of exotic spheres
• Donaldson's disproof of the h-cobordism theorem in dimension 4
  • Smooth structures on 4-manifolds more complicated than higher dimensions
• Inspired development of Morse-Floer theory and topological quantum field theories
  • Morse homology, Floer homology, Morse-Witten complex

Related Theorems and Extensions

• s-cobordism theorem strengthens h-cobordism theorem with extra conditions
  • Whitehead torsion of the inclusion maps vanishes
• Barden-Mazur-Stallings theorem classifies h-cobordisms in dimension 5
  • Not all h-cobordisms are trivial, obstructions lie in certain nilpotent groups
• Kervaire-Milnor classification of exotic spheres in dimensions $n \geq 5$
  • Uses h-cobordism theorem and surgery theory
• Smale's proof of the generalized Poincaré conjecture in dimensions $n \geq 5$
  • Consequence of the h-cobordism theorem
• Morse homology a powerful generalization of Morse theory
  • Chain complex generated by critical points, differential counts gradient trajectories
• Cerf theory studies families of Morse functions and their singularities
  • Important in the study of diffeomorphism groups of manifolds

Exercises and Problem-Solving Techniques

• Compute the Morse homology of the torus $T^2$ with different Morse functions
  • Relate to the standard cell decomposition and homology groups
• Show that the Morse inequalities are sharp for complex projective spaces $\mathbb{CP}^n$
  • Find a Morse function with the minimum number of critical points
• Prove the Reeb theorem using Morse theory
  • Every compact manifold admits a Morse function with exactly two critical points
• Analyze the topology of sublevel sets for the height function on the 2-sphere
  • Describe the attachment of handles and changes in homology
• Construct an exotic 7-sphere using the h-cobordism theorem and surgery
  • Start with a homotopy 7-sphere and perform suitable surgeries
• Prove the h-cobordism theorem in dimension 6 using Smale's cancellation theorem
  • Rearrange and cancel handles to trivialize the cobordism
• Find a Morse function on the 3-torus $T^3$ with the minimum number of critical points
  • Relate the critical points to a cell decomposition of $T^3$