10.2 Proof and implications of Morse inequalities

The Morse inequalities are a powerful tool in Morse theory, connecting the critical points of a Morse function to the topology of a manifold. They provide a way to estimate the Betti numbers of a manifold based on the number of critical points of different indices.

This section dives into the proof of the Morse inequalities and explores their implications. We'll see how these inequalities relate the Morse and Poincaré polynomials, giving us insights into the manifold's topology through its critical points.

Morse Complex and Homology

Morse Complex and Boundary Operator

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• Morse complex is a chain complex constructed from the critical points of a Morse function on a manifold
• Each critical point of index $k$ generates a basis element for the $k$-th vector space in the complex
• Boundary operator $\partial_k$ maps from the $k$-th vector space to the $(k-1)$-th vector space in the complex
• Boundary operator captures the connectivity between critical points of adjacent indices (index $k$ and $k-1$)
• Boundary operator satisfies the property $\partial_{k-1} \circ \partial_k = 0$ for all $k$, meaning the composition of two consecutive boundary maps is always zero

Homology Groups and Exact Sequence

• Homology groups measure the "holes" in a topological space captured by the Morse complex
• $k$-th homology group $H_k$ is defined as the quotient $\ker(\partial_k) / \operatorname{im}(\partial_{k+1})$
• Elements of $H_k$ are equivalence classes of cycles (elements in $\ker(\partial_k)$) modulo boundaries (elements in $\operatorname{im}(\partial_{k+1})$)
• Rank of $H_k$ counts the number of independent "holes" of dimension $k$ in the space
• Morse complex gives rise to a long exact sequence of homology groups connecting the critical points of different indices
• Exact sequence provides a relationship between the homology groups of the Morse complex and the original manifold

Algebraic Topology Tools

Rank-Nullity Theorem and Induction

• Rank-nullity theorem states that for a linear map $f: V \to W$ between vector spaces, $\dim(V) = \operatorname{rank}(f) + \operatorname{nullity}(f)$
• In the context of Morse theory, rank-nullity theorem relates the dimensions of the vector spaces in the Morse complex
• Induction is a proof technique used to establish a statement for all natural numbers
• Induction starts with a base case and then proves that if the statement holds for $n$, it must also hold for $n+1$
• Induction is often used to prove statements about the Morse complex and its homology groups across all dimensions

Poincaré and Morse Polynomials

• Poincaré polynomial $P(t)$ is a generating function that encodes the ranks of the homology groups of a space
• Coefficient of $t^k$ in $P(t)$ equals the rank of the $k$-th homology group $H_k$
• Poincaré polynomial provides a compact way to express the homology of a space (e.g., $P(t) = 1 + 2t + t^2$ for a torus)
• Morse polynomial $M(t)$ is a generating function that encodes the number of critical points of each index for a Morse function
• Coefficient of $t^k$ in $M(t)$ equals the number of critical points of index $k$
• Morse polynomial is related to the Poincaré polynomial via the Morse inequalities, which compare the coefficients of the two polynomials