12.1 Morse functions and critical points

Morse functions are smooth real-valued functions on manifolds with non-degenerate critical points. They're key tools in studying manifold topology, revealing how the structure changes as we move through different function values.

Critical points of Morse functions are where the fun happens. These points, classified by their index, show where the topology shifts. As we pass through them, we see handles attaching and the manifold's shape evolving.

Morse functions and their properties

Definition and properties of Morse functions

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• A Morse function is a smooth real-valued function on a smooth manifold whose critical points are non-degenerate
  • Non-degenerate critical points are points where the Hessian matrix of second partial derivatives is non-singular
  • The Hessian matrix is the matrix of second partial derivatives of the function
  • Non-singularity means that the determinant of the Hessian matrix is non-zero
• Morse functions are generic in the space of smooth functions
  • A small perturbation of any smooth function yields a Morse function
  • This implies that Morse functions are abundant and can be easily constructed
• The Morse Lemma states that near a non-degenerate critical point, a Morse function has a canonical form up to a change of coordinates
  • The canonical form is a quadratic function with a specific number of positive and negative terms
  • The change of coordinates is a local diffeomorphism that simplifies the function near the critical point

Sublevel sets and handle attachments

• The sublevel sets of a Morse function undergo topological changes when passing through a critical point
  • A sublevel set is the set of points where the function value is less than or equal to a given value
  • Passing through a critical point of index $k$ corresponds to attaching a $k$-dimensional handle to the sublevel set
  • A handle is a product of a $k$-dimensional disc and an $(n-k)$-dimensional disc, where $n$ is the dimension of the manifold
• The change in topology near a critical point can be described by a handle attachment or a surgery operation
  • Surgery is the process of removing a neighborhood of a submanifold and replacing it with another submanifold with the same boundary
  • Handle attachments and surgeries provide a way to construct and modify manifolds using Morse functions

Critical points and their index

Definition and classification of critical points

• Critical points of a Morse function are points where the gradient vanishes
  • The gradient is the vector of partial derivatives of the function
  • At a critical point, all partial derivatives are zero
• The Hessian matrix at a critical point determines the type of the critical point
  • The index of a critical point is the number of negative eigenvalues of the Hessian matrix
  • Eigenvalues are the roots of the characteristic polynomial of the Hessian matrix
• Critical points can be classified as minima (index 0), saddles (index between 1 and $n-1$), or maxima (index $n$)
  • Minima are critical points with no negative eigenvalues (local minimum)
  • Saddles have both positive and negative eigenvalues (neither local minimum nor maximum)
  • Maxima have all negative eigenvalues (local maximum)

Morse index theorem and its implications

• The Morse index theorem relates the index of a critical point to the topology of the sublevel sets of the Morse function
  • It states that the index of a critical point is equal to the number of negative eigenvalues of the Hessian matrix
  • The theorem provides a link between the local behavior of the function near a critical point and the global topology of the manifold
• The index of a critical point determines the homology of the sublevel sets
  • The homology groups measure the number of holes or cycles in different dimensions
  • The index of a critical point is related to the change in homology when passing through that critical point
  • Critical points of index $k$ create or destroy $k$-dimensional holes in the sublevel sets

Topology of manifolds and critical points

Morse inequalities and Betti numbers

• The critical points of a Morse function provide information about the topology of the manifold
  • The number and type of critical points are related to the homology and Betti numbers of the manifold
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
  • Betti numbers are topological invariants that count the number of independent holes or cycles in each dimension
  • The weak Morse inequalities state that the number of critical points of index $k$ is greater than or equal to the $k$-th Betti number
  • The strong Morse inequalities involve alternating sums of critical points and Betti numbers
• The Morse inequalities provide lower bounds for the number of critical points based on the topology of the manifold
  • They can be used to estimate the complexity of a Morse function and the topology of the manifold
  • The inequalities are sharp for certain classes of manifolds, such as spheres and projective spaces

Morse-Smale complex and cellular decomposition

• The Morse-Smale complex is a cellular decomposition of the manifold based on the flow lines connecting critical points
  • A cellular decomposition is a partition of the manifold into cells of different dimensions
  • Flow lines are integral curves of the gradient vector field of the Morse function
  • The Morse-Smale complex is constructed by considering the stable and unstable manifolds of critical points
• The cells of the Morse-Smale complex are determined by the critical points and their indices
  • Each critical point of index $k$ corresponds to a $k$-dimensional cell in the complex
  • The boundary of a cell consists of lower-dimensional cells corresponding to critical points of lower index
• The Morse-Smale complex provides a way to study the global structure of the manifold using the local information at critical points
  • It encodes the connectivity and adjacency relations between critical points
  • The homology of the manifold can be computed from the Morse-Smale complex using cellular homology

Morse function behavior near critical points

Morse Lemma and local canonical form

• The Morse Lemma provides a local canonical form for a Morse function near a critical point
  • It states that in a neighborhood of a non-degenerate critical point, the Morse function can be expressed as a quadratic form
  • The quadratic form has a specific number of positive and negative terms determined by the index of the critical point
• Near a critical point of index $k$, the Morse function has a quadratic form with $k$ negative terms and $(n-k)$ positive terms
  • The quadratic form is obtained by a change of coordinates that diagonalizes the Hessian matrix
  • The canonical form simplifies the local behavior of the function and allows for a standard description of the critical point

Topological changes and handle attachments

• The sublevel sets of a Morse function undergo topological changes when passing through a critical point
  • The type of topological change depends on the index of the critical point
  • Passing through a critical point of index $k$ corresponds to attaching a $k$-dimensional handle to the sublevel set
• Handle attachments provide a way to describe the change in topology near a critical point
  • A handle of index $k$ is a product of a $k$-dimensional disc and an $(n-k)$-dimensional disc
  • Attaching a handle of index $k$ to a sublevel set means gluing the boundary of the $k$-dimensional disc to the boundary of the sublevel set
• The effect of a handle attachment on the homology of the sublevel set can be determined by the Morse index theorem
  • Attaching a handle of index $k$ changes the homology in dimension $k$ and $k-1$
  • The change in homology is related to the creation or destruction of $k$-dimensional holes in the sublevel set