Continuation maps for the Morse fundamental group

TL;DR

The paper constructs and analyzes continuation maps for the Morse fundamental group $\pi_1^{\mathrm{Morse}}(f,\ast)$, showing how interpolation data $(F,G)$ induce morphisms between Morse fundamental groups and proving their functoriality. It extends the construction to grafted trajectories, enabling comparisons of Morse fundamental groups across different manifolds and establishing isotopy invariance via canonical isomorphisms. A detailed treatment of interpolation-type functions on $M\times\mathbb{R}$ yields a relative fundamental-group interpretation: $\pi_1^{\mathrm{Morse}}(F,\ast)$ is isomorphic to $\pi_1(\mathbb{R}\times M, F^{-1}(-\infty,c]\cup\{\ast'\},\ast')$, linking Morse-theoretic data to relative/topological frameworks. Together, these results provide a robust, Morse-theoretic approach to fundamental-group invariants, invariance under data choices, and a pathway to relative constructions in noncompact settings.

Abstract

We study properties of the continuation map for the Morse fundamental group $π_1^\text{Morse}(f,\ast)$ associated to a Morse-Smale pair $(f,g)$ on a manifold $M$. We get a morphism between $π_1^\text{Morse}(f_1,\ast_1)$ and $π_1^\text{Morse}(f_2,\ast_2)$ and show that it is functorial. We also define the morphism in the case of Morse data over different manifolds, thanks to the use of grafted trajectories. Finally, given an interpolation function on $M\times\mathbb{R}$ between two Morse functions (used for example to define the continuation map), we study the Morse fundamental group associated to that function and show that it is isomorphic to a relative fundamental group on $M\times\mathbb{R}$.

Figures (14)

• Figure 1: $F: M\times\mathbb{R}\rightarrow\mathbb{R}$ which has a maximum at $s=0$ in $f_1$ and minimum at $s=1$ in $f_2$.
• Figure 2: Positive side of the unstable manifold of $(x,0) \in {\rm Crit}_2(F)$: the image of the step formed by $(\beta_1,\beta_2)$ by $\tilde{\phi}$ is the concatenation of $f_2$-steps which goes from $(m_1,1)$ to $(m_2,1)$.
• Figure 3: The image of the moduli space $\overline{\mathcal{M}_+((z,0),\varnothing)}$ by ev for $z \in {\rm Crit}_2(f_1)$, with one end in $M\times\{0\}$ formed by the unstable manifold of $z$, and one end in $M\times\{1\}$ formed by the union of the unstable manifolds of $z'_1, z'_2, z'_3$ and $x'$. The unstable manifolds of each point of the form $(x,0)$, where $x\in{\rm Crit}_1(f_1)$ is a point in the unstable manifold of $z$, are in color. We may read the image of each step through these points by $\phi_F$ on the other side of the colored zone (the two dotted lines which bound the purple zone are identified together).
• Figure 4: On the left, a schematic illustration of $\mathcal{M}((z,0),M\times\{1\})$ formed of moduli spaces which correspond to breaks in index 2 critical points $z_1$ and $z_2$ (in grey, with the rigid trajectories from $z_1$ and $z_2$ depicted by dotted red lines), separated by moduli spaces which correspond to breaks in index 1 (in blue) or 0 (in yellow) critical point. On the right, the image of $\mathcal{M}((z,0),M\times\{1\})$ by ${\rm ev}_{f_2}$: the blue zones are sent onto lines and the yellow zones sent onto points (and in particular, the boundary parts of the grey zones which touch the yellow zones also are sent onto points), all of which together cover the boundaries of the unstable manifolds of $z_1$ and $z_2$.
• Figure 5: On the left, a schematic illustration of two strips (in red and blue) in $\mathcal{M}((z,0),M\times\{1\})$ which correspond to moduli spaces of broken trajectories at index 1 critical points which go from one part of the boundary to another (surrounded by yellow strips which correspond to the moduli spaces of the form $\mathcal{M}((z,1),(y,0))\times\mathcal{M}((y,0),\varnothing)$, where each $y$ is an index 0 critical point in the unstable manifolds of the index 1 critical points). On the right, the image of $\mathcal{M}((z,0),M\times\{1\})$ by ${\rm ev}_{f_2}$.

Theorems & Definitions (54)

• Definition 1.1
• Definition 1.2
• Remark 1.1
• Theorem : folklore
• Lemma 1.1
• proof
• Remark 1.2
• Theorem 1.1
• Theorem 1.2
• Corollary 1.1