12.4 Morse homology and the Morse-Witten complex

Morse homology connects the critical points of a Morse function to a manifold's topology. By analyzing gradient flows between these points, we can construct the Morse-Witten complex, whose homology mirrors the manifold's singular homology.

This powerful tool simplifies topological computations and yields insights like Morse inequalities. It bridges differential geometry and algebraic topology, offering a concrete way to understand a manifold's shape through its critical points and flows.

Morse-Witten Complex Definition

Construction from a Morse Function

• The Morse-Witten complex is a chain complex constructed from a Morse function on a smooth manifold
• A Morse function is a smooth real-valued function on a manifold with non-degenerate critical points (local minima, maxima, and saddle points)
• The chain groups of the Morse-Witten complex are generated by the critical points of the Morse function, graded by their Morse index (number of negative eigenvalues of the Hessian matrix at the critical point)
• The differential in the Morse-Witten complex is defined by counting the gradient flow lines between critical points of consecutive indices
  • The gradient flow of a Morse function is a one-parameter group of diffeomorphisms obtained by integrating the negative gradient vector field
  • The differential counts the number of gradient flow lines (modulo 2) between critical points, satisfying certain compactness and transversality conditions
  • Example: For a height function on a torus, the differential counts the number of gradient flow lines between a saddle point and a minimum or maximum

Dependence on Additional Structures

• The construction of the Morse-Witten complex depends on the choice of the Morse function and an additional metric on the manifold
• Different Morse functions on the same manifold may yield different Morse-Witten complexes, but their homology groups are isomorphic
• The metric on the manifold is used to define the gradient vector field and the inner product structure on the tangent spaces
• The compactness and transversality conditions for the gradient flow lines may require perturbing the Morse function or the metric
• Example: On a sphere, the height function with respect to different axes yields different Morse-Witten complexes, but they all compute the homology of the sphere

Gradient Flows in Morse Homology

Connecting Critical Points

• Gradient flows connect critical points of a Morse function and determine the differential in the Morse-Witten complex
• The gradient vector field of a Morse function points in the direction of steepest ascent and is orthogonal to the level sets of the function
• The gradient flow lines are integral curves of the negative gradient vector field, flowing from higher to lower critical points
• Example: On a surface, gradient flow lines connect saddle points to minima and maxima, creating a cellular decomposition of the surface

Compactness and Transversality

• The compactness of the moduli spaces of gradient flow lines is crucial for the well-definedness of the differential in the Morse-Witten complex
  • Compactness is achieved by imposing suitable boundedness and convergence conditions on the gradient flow lines
  • Example: For a compact manifold, the space of gradient flow lines connecting two critical points is compact, ensuring a finite count
• The transversality of the stable and unstable manifolds of critical points ensures that the gradient flow lines are isolated and can be counted
  • The stable manifold of a critical point consists of points that flow to the critical point under the gradient flow
  • The unstable manifold of a critical point consists of points that flow from the critical point under the gradient flow
  • Transversality means that the stable and unstable manifolds intersect in a generic way, with the expected dimension
• The Morse-Smale condition, which requires the stable and unstable manifolds to intersect transversely, is a sufficient condition for defining the Morse-Witten complex

Homology of the Morse-Witten Complex

Morse Homology Theorem

• The homology of the Morse-Witten complex is defined as the quotient of the kernel of the differential by its image
• The Morse Homology Theorem states that the homology of the Morse-Witten complex is isomorphic to the singular homology of the manifold
  • This isomorphism is induced by a chain map from the Morse-Witten complex to the singular chain complex, which sends each critical point to a sum of nearby singular simplices
  • Example: For the height function on a torus, the Morse homology groups are isomorphic to the singular homology groups of the torus

Morse Inequalities

• The Morse inequalities relate the Betti numbers of the manifold to the number of critical points of each index in a Morse function
  • The weak Morse inequalities state that the Betti numbers are bounded above by the number of critical points of the corresponding index
  • The strong Morse inequalities involve an alternating sum of the number of critical points and provide a more refined estimate of the Betti numbers
  • Example: For a surface of genus g, a Morse function has at least 2g+2 critical points, with at least one minimum, one maximum, and 2g saddle points
• The Morse-Witten complex provides a way to compute the homology of a manifold using the critical points of a Morse function, which can be more efficient than using singular homology

Morse Homology for Topology Analysis

Topological Information from Critical Points

• Morse homology can be used to derive topological information about a manifold from the critical points of a Morse function
• The Euler characteristic of a manifold can be expressed as the alternating sum of the number of critical points of a Morse function
• The Poincaré polynomial of a manifold, which encodes its Betti numbers, can be bounded using the Morse inequalities
• Example: The Euler characteristic of a torus is 0, which can be seen from a Morse function with one minimum, one maximum, and two saddle points

Topological Invariance and Applications

• Morse homology is independent of the choice of the Morse function and the metric on the manifold, making it a topological invariant
  • Different Morse functions on the same manifold yield isomorphic Morse homology groups
  • Homotopy equivalent manifolds have isomorphic Morse homology groups
• The Morse Homology Theorem implies that Morse homology is a stronger invariant than singular homology, as it captures additional information about the gradient flow
• Morse homology can be used to prove the h-cobordism theorem, which characterizes the topology of cobordisms between manifolds in terms of their homology
  • A cobordism is a manifold with boundary that interpolates between two given manifolds
  • The h-cobordism theorem states that if the cobordism has trivial homology relative to its boundary, then it is diffeomorphic to a product cobordism
• The functoriality of Morse homology with respect to smooth maps between manifolds allows for the study of induced homomorphisms and the construction of long exact sequences
  • A smooth map between manifolds induces a chain map between their Morse-Witten complexes, which in turn induces a homomorphism between their Morse homology groups
  • Long exact sequences in Morse homology can be used to study the relationship between the topology of a manifold and its submanifolds or quotient spaces