7.2 Cellular homology and Morse theory

Cellular homology and Morse theory are powerful tools for understanding the topology of spaces. They connect geometric structures to algebraic computations, revealing how a space's shape relates to its homology groups.

These techniques allow us to study complex spaces by breaking them into simpler pieces. By analyzing critical points and gradient flows, we can extract key information about a space's topology and compute important invariants like Betti numbers.

Cellular Homology and Chain Complexes

Cellular Decomposition and Homology

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• Cellular homology computes the homology groups of a cell complex using the boundary information of its cells
• Decomposes a topological space into a collection of cells (vertices, edges, faces, etc.) glued together in a specific way
• Captures the connectivity and "holes" in the space by studying the incidence relations between cells
• Provides a combinatorial approach to compute topological invariants such as Betti numbers and Euler characteristic

Chain Complexes and Boundary Operators

• Chain complex is an algebraic structure consisting of a sequence of abelian groups (chain groups) connected by homomorphisms (boundary operators)
  • Each chain group $C_n$ represents the group of n-dimensional cells in the complex
  • Boundary operators $\partial_n: C_n \rightarrow C_{n-1}$ encode the incidence relations between cells of adjacent dimensions
• Boundary operator satisfies the property $\partial_{n-1} \circ \partial_n = 0$, meaning the composition of two consecutive boundary maps is always zero
  • This property allows the definition of homology groups as quotients of kernel and image subgroups
• Homology groups $H_n(C_*) = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})$ measure the "holes" in the complex that are not boundaries of higher-dimensional cells

Euler Characteristic and Homology

• Euler characteristic $\chi$ is a topological invariant that can be computed from a cell complex as the alternating sum of the number of cells in each dimension
  • $\chi = \sum_{n} (-1)^n c_n$, where $c_n$ is the number of n-dimensional cells
• Euler characteristic is also related to the Betti numbers $\beta_n$ (ranks of homology groups) by the formula $\chi = \sum_{n} (-1)^n \beta_n$
  • This relationship connects the combinatorial data of the cell complex with the algebraic data of homology groups
• Examples of Euler characteristic:
  • For a sphere, $\chi = 2$ (2 cells - 0 cells)
  • For a torus, $\chi = 0$ (1 cell - 2 cells + 1 cell)

Morse Theory Fundamentals

Morse Functions and Critical Points

• Morse function is a smooth real-valued function on a manifold whose critical points are non-degenerate (Hessian matrix is non-singular)
  • Critical points are points where the gradient of the function vanishes
  • Non-degenerate critical points are isolated and have a well-defined index (number of negative eigenvalues of the Hessian)
• Morse functions provide a way to study the topology of a manifold through the configuration of their critical points
  • The index of a critical point determines the type of topological change that occurs at that level (birth or death of a homology class)
• Examples of Morse functions:
  • Height function on a torus has 4 critical points: 1 minimum, 2 saddles, 1 maximum
  • Height function on a sphere has 2 critical points: 1 minimum, 1 maximum

Morse-Smale Transversality and Gradient Flow

• Morse-Smale transversality condition requires that the stable and unstable manifolds of critical points intersect transversely
  • Stable manifold of a critical point consists of points that flow to the critical point under the gradient flow
  • Unstable manifold consists of points that flow away from the critical point
• Gradient flow is a vector field on the manifold defined as the negative gradient of the Morse function
  • Integral curves of the gradient flow connect critical points and provide a way to decompose the manifold into cells
• Morse complex is a cellular decomposition of the manifold obtained from the gradient flow, where each cell corresponds to a critical point
  • The dimension of the cell is equal to the index of the corresponding critical point
  • The boundary of a cell consists of lower-dimensional cells corresponding to critical points connected by gradient flow lines

Morse Inequalities and Homology

• Morse inequalities relate the number of critical points of a Morse function to the Betti numbers of the manifold
  • The weak Morse inequalities state that $m_k \geq \beta_k$, where $m_k$ is the number of critical points of index $k$
  • The strong Morse inequalities provide more refined bounds on the alternating sums of Betti numbers and critical points
• Morse homology is an approach to compute the homology groups of a manifold using the Morse complex
  • The boundary operator in Morse homology counts the gradient flow lines between critical points
  • Morse homology is isomorphic to the singular homology of the manifold, providing a link between the analytical data of the Morse function and the topological invariants

Topological Invariants

Betti Numbers and Homology Groups

• Betti numbers $\beta_k$ are topological invariants that measure the number of independent k-dimensional "holes" in a topological space
  • $\beta_0$ counts the number of connected components
  • $\beta_1$ counts the number of 1-dimensional holes (loops)
  • $\beta_2$ counts the number of 2-dimensional voids, and so on
• Homology groups $H_k(X)$ are algebraic objects that generalize the notion of Betti numbers and provide more refined information about the holes
  • Elements of $H_k(X)$ are equivalence classes of k-dimensional cycles (subspaces without boundary) modulo boundaries of (k+1)-dimensional subspaces
  • The rank of $H_k(X)$ is equal to the Betti number $\beta_k$
• Examples of Betti numbers and homology groups:
  • For a circle, $\beta_0 = 1$ (connected), $\beta_1 = 1$ (one loop), and $H_1(S^1) \cong \mathbb{Z}$
  • For a torus, $\beta_0 = 1$, $\beta_1 = 2$ (two independent loops), $\beta_2 = 1$ (one void), and $H_1(T^2) \cong \mathbb{Z}^2$
• Betti numbers and homology groups are important tools in algebraic topology for distinguishing topological spaces and studying their properties
  • They are invariant under continuous deformations (homeomorphisms) and capture intrinsic features of the space
  • Computations of these invariants often involve techniques from abstract algebra, such as exact sequences and commutative diagrams