7.1 CW complex structure from Morse functions

Morse functions help us understand a manifold's shape by looking at its critical points. These functions create a CW complex structure, breaking the manifold into simpler pieces we can study.

By examining how cells attach to each other, we can figure out the manifold's topology. This method connects Morse Theory to the broader study of manifold structure and classification.

CW Complex Structure

Cell Attachment and CW Complexes

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• CW complex constructed by iteratively attaching cells of increasing dimension
• Start with a discrete set of points called 0-cells
• Inductively attach $n$-cells to the $(n-1)$-skeleton via continuous maps from the boundary of an $n$-disk to the $(n-1)$-skeleton
• The attachment of an $n$-cell is determined by a continuous map $\phi: S^{n-1} \to X^{(n-1)}$ where $S^{n-1}$ is the boundary of the $n$-disk and $X^{(n-1)}$ is the $(n-1)$-skeleton
• The resulting space after attaching all cells is a CW complex

Properties of CW Complexes

• Each point in a CW complex belongs to the interior of a unique cell (open cell)
• The closure of each cell is homeomorphic to a closed disk
• The boundary of each cell is contained in the union of lower-dimensional cells
• CW complexes provide a convenient way to build spaces by successively attaching cells of increasing dimension (skeleton structure)

Attaching Maps and Cell Structure

• Attaching maps $\phi: S^{n-1} \to X^{(n-1)}$ specify how $n$-cells are glued to the $(n-1)$-skeleton
• The choice of attaching maps determines the topological structure of the CW complex
• Different attaching maps can result in different topological spaces (torus, projective plane)
• The cell structure of a CW complex encodes important topological information about the space
• Cellular homology can be computed using the cell structure and boundary maps induced by the attaching maps

Morse Theory Fundamentals

Critical Points and Their Indices

• Morse function $f: M \to \mathbb{R}$ assigns a real number to each point on a smooth manifold $M$
• Critical points of $f$ are points where the gradient $\nabla f$ vanishes
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix of $f$ at that point
• Critical points are classified as minima (index 0), saddles (index 1 to $n-1$), and maxima (index $n$) where $n$ is the dimension of $M$
• The index of a critical point determines the type of topological change that occurs at that level set of $f$

Descending Manifolds and Flow Lines

• For each critical point $p$ of index $k$, there is an associated descending manifold $D(p)$ of dimension $n-k$
• Descending manifolds are submanifolds of $M$ consisting of points that flow to $p$ under the negative gradient flow of $f$
• The boundary of $D(p)$ is contained in the union of descending manifolds of critical points of index greater than $k$
• Flow lines are integral curves of the negative gradient vector field $-\nabla f$
• Flow lines connect critical points and provide a way to visualize the topology of the manifold (gradient flow)

Morse Functions and Topology

• Morse functions encode the topology of the manifold through their critical points and descending manifolds
• The number and indices of critical points are related to the Betti numbers and Euler characteristic of the manifold (Morse inequalities)
• Changes in topology occur at critical levels of the Morse function (level sets)
• Morse theory establishes a correspondence between the critical points of a Morse function and the handles attached to the manifold (handle decomposition)

Morse-Smale Complex

Definition and Properties

• Morse-Smale complex is a cellular decomposition of a manifold $M$ induced by a Morse-Smale function $f$
• A Morse-Smale function is a Morse function whose ascending and descending manifolds intersect transversely
• The cells of the Morse-Smale complex are the connected components of the intersections of ascending and descending manifolds
• Each cell is labeled by a pair of critical points $(p, q)$ where $p$ and $q$ are the unique minimum and maximum of $f$ on the cell respectively
• The dimension of a cell labeled $(p, q)$ is $\text{index}(q) - \text{index}(p)$

Handlebody Decomposition and Topology

• The Morse-Smale complex provides a handlebody decomposition of the manifold $M$
• Each critical point of index $k$ corresponds to the attachment of a $k$-handle to the boundary of the handlebody
• The attaching maps of the handles are determined by the descending manifolds of the critical points
• The handlebody decomposition encodes the topology of the manifold (homology, homotopy type)
• The Morse-Smale complex can be used to compute topological invariants such as Betti numbers and the Euler characteristic
• The combinatorial structure of the Morse-Smale complex captures the essential topological features of the manifold (simplification, visualization)