[Paper Review] Topology of the spaces of Morse functions on surfaces

Elena Kudryavtseva|arXiv (Cornell University)|Apr 25, 2011

Topological and Geometric Data Analysis3 citations

TL;DR

This paper establishes a homotopy equivalence between the space $ F $ of Morse functions on a closed oriented surface $ M $, with at least $ \chi(M)+1 $ labeled critical points, and the product $ R \times \widetilde{\mathcal{M}} $, where $ R $ is a manifold depending on the Euler characteristic $ \chi(M) $ (e.g., $ \mathbb{R}P^3 $, $ S^1 \times S^1 $, or a point), and $ \widetilde{\mathcal{M}} $ is the universal moduli space of framed Morse functions. The result reduces the homotopical study of $ F $ to the combinatorial and stratified topology of $ \widetilde{\mathcal{M}} $, with Morse-type inequalities derived for its Betti numbers.

ABSTRACT

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $χ(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R imes{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1 imes S^1$ and the point in dependence on the sign of $χ(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.

Motivation & Objective

• To determine the homotopy type of the space $ F $ of Morse functions on a closed oriented surface $ M $ with at least $ \chi(M)+1 $ labeled critical points.
• To reduce the topological complexity of $ F $ to the study of the universal moduli space $ \widetilde{\mathcal{M}} $ of framed Morse functions.
• To derive Morse-type inequalities for the Betti numbers of $ F $ using the stratified structure of $ \widetilde{\mathcal{M}} $.
• To establish a homotopy equivalence $ F \sim R \times \widetilde{\mathcal{M}} $, where $ R $ depends on the sign of $ \chi(M) $.

Proposed method

• Use of $ C^\infty $-topology on the space $ F $ of Morse functions with labeled critical points.
• Construction of the universal moduli space $ \widetilde{\mathcal{M}} $ as a smooth stratified manifold via framed Morse functions.
• Application of the homotopy equivalence $ F \sim \mathbb{F} $, where $ \mathbb{F} $ is the space of framed Morse functions, established in prior work.
• Proof of a homeomorphism $ \mathbb{F}^1 \approx \mathscr{D}^0 \times \widetilde{\mathcal{M}} $, where $ \mathbb{F}^1 $ is a subspace of framed functions with fixed labeling.
• Use of natural stratification and induced neighborhood stratification on $ \widetilde{\mathcal{M}} $ to analyze its homology and Betti numbers.
• Derivation of Morse inequalities for Betti numbers of $ F $ via the structure of $ \widetilde{\mathcal{M}} $, particularly in the case $ M = S^2 $.

Experimental results

Research questions

• RQ1 What is the homotopy type of the space $ F $ of Morse functions on a closed oriented surface $ M $ with at least $ \chi(M)+1 $ labeled critical points?
• RQ2 How does the topology of $ F $ decompose in terms of simpler topological invariants?
• RQ3 What are the Betti number inequalities for $ F $, and how are they derived from the structure of the moduli space $ \widetilde{\mathcal{M}} $?
• RQ4 How does the Euler characteristic of $ \widetilde{\mathcal{M}} $ relate to the topology of $ M $, especially for $ M = S^2 $?

Key findings

• The space $ F $ of Morse functions on a closed oriented surface $ M $ with at least $ \chi(M)+1 $ labeled critical points is homotopy equivalent to $ R \times \widetilde{\mathcal{M}} $, where $ R $ is $ \mathbb{R}P^3 $, $ S^1 \times S^1 $, or a point depending on the sign of $ \chi(M) $.
• The universal moduli space $ \widetilde{\mathcal{M}} $ of framed Morse functions is a smooth stratified manifold of dimension $ 3q $, where $ q $ is the number of saddle points.
• For $ M = S^2 $, Morse inequalities for the Betti numbers of $ \widetilde{\mathcal{M}} $ are derived, and its Euler characteristic is computed.
• The space $ \mathbb{F}^1 $ of framed functions with fixed labeling is homeomorphic to $ \mathscr{D}^0 \times \widetilde{\mathcal{M}} $, where $ \mathscr{D}^0 $ is the identity component of the diffeomorphism group.
• The inclusion $ F^1 \hookrightarrow F $ and the forgetful map $ \mathbb{F} \to F $ are homotopy equivalences, justifying the reduction to $ \widetilde{\mathcal{M}} $.
• The result provides a complete topological reduction of $ F $ to the combinatorial and stratified topology of $ \widetilde{\mathcal{M}} $, enabling further homological analysis.