4.3 Relationship between index and local topology

The index of a critical point in Morse theory reveals crucial information about the local topology of a manifold. It determines how the manifold's shape changes near that point, affecting the overall structure.

Understanding the relationship between index and local topology helps us grasp how critical points contribute to a manifold's global shape. This connection is key to using Morse theory for analyzing and classifying manifolds.

Local Structure of Critical Points

Morse Lemma and Local Diffeomorphism

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• The Morse Lemma states that near a non-degenerate critical point $p$ of a smooth function $f$, there exists a coordinate system $(x_1, \ldots, x_n)$ in which $f$ takes the form $f(x) = f(p) - x_1^2 - \ldots - x_{\lambda}^2 + x_{\lambda+1}^2 + \ldots + x_n^2$
  • $\lambda$ is the index of the critical point $p$
  • This coordinate system is obtained through a local diffeomorphism (a smooth bijective map with a smooth inverse) around $p$
• The Morse Lemma implies that the local behavior of a function near a non-degenerate critical point is completely determined by the index of the critical point
  • The index $\lambda$ counts the number of independent directions in which $f$ decreases from $p$
• The local diffeomorphism guarantees that the function $f$ has the same qualitative behavior as the quadratic form $-x_1^2 - \ldots - x_{\lambda}^2 + x_{\lambda+1}^2 + \ldots + x_n^2$ near the critical point $p$

Handle Attachment and CW Complexes

• The local structure of a critical point can be understood through the process of handle attachment
  • An $n$-dimensional $\lambda$-handle is a product $D^\lambda \times D^{n-\lambda}$, where $D^k$ denotes the $k$-dimensional disk
  • Attaching a $\lambda$-handle to a manifold $M$ corresponds to gluing $D^\lambda \times D^{n-\lambda}$ to $M$ along $\partial D^\lambda \times D^{n-\lambda}$ (the boundary of the $\lambda$-disk times the $(n-\lambda)$-disk)
• The attachment of handles to a manifold can be used to build a CW complex, a topological space constructed by inductively attaching cells of increasing dimension
  • The local structure near a critical point of index $\lambda$ corresponds to the attachment of a $\lambda$-handle
  • The resulting CW complex captures the topology of the manifold and provides a way to understand the relationship between critical points and the global topology

Topology of Level Sets

Homology Groups and Poincaré Polynomial

• The topology of the level sets $f^{-1}(a)$ of a Morse function $f$ can be studied using homology groups
  • Homology groups $H_k(X)$ measure the "holes" in a topological space $X$ in each dimension $k$ (e.g., connected components in dimension 0, loops in dimension 1, voids in dimension 2)
  • The rank of the homology group $H_k(X)$ is called the $k$-th Betti number, denoted by $\beta_k(X)$
• The Poincaré polynomial of a topological space $X$ is the generating function of its Betti numbers: $P_X(t) = \sum_{k=0}^{\infty} \beta_k(X) t^k$
  • The Poincaré polynomial encodes the ranks of the homology groups of $X$ in a compact form
  • For a compact manifold $M$, the Poincaré polynomial is a polynomial (finite sum) because $H_k(M) = 0$ for $k > \dim(M)$

Euler Characteristic and Morse Inequalities

• The Euler characteristic of a topological space $X$ is the alternating sum of its Betti numbers: $\chi(X) = \sum_{k=0}^{\infty} (-1)^k \beta_k(X)$
  • For a compact manifold $M$, the Euler characteristic can be computed as $\chi(M) = \sum_{k=0}^{\dim(M)} (-1)^k \beta_k(M)$
  • The Euler characteristic is a topological invariant, meaning that it is preserved under homeomorphisms (continuous bijections with continuous inverses)
• The Morse inequalities relate the Betti numbers of a compact manifold $M$ to the number of critical points of a Morse function $f$ on $M$
  • Let $c_k$ denote the number of critical points of $f$ with index $k$. Then, for each $k$, we have: $\beta_k(M) \leq c_k$
  • Moreover, the alternating sum of the Betti numbers equals the alternating sum of the numbers of critical points: $\chi(M) = \sum_{k=0}^{\dim(M)} (-1)^k \beta_k(M) = \sum_{k=0}^{\dim(M)} (-1)^k c_k$
• The Morse inequalities provide a powerful tool for studying the topology of a manifold using a Morse function defined on it
  • They give lower bounds on the number of critical points of each index that any Morse function must have
  • In particular, the Euler characteristic can be computed by counting the critical points of a Morse function with appropriate signs based on their indices