Fundamental group in stable Morse theory

Abstract

Morse theory relates algebraic topology invariants and the dynamics of the gradient flow of a Morse function, allowing to derive information about one out of the other. In the case of the homology, the construction extends to much more general settings, and in particular to the infinite dimensional setting of the celebrated Floer homology in symplectic geometry. The case of the fundamental group is quiet different however, and the object of this paper is to provide a dynamical description of the fundamental group in the stable Morse setting, which can be thought of as an intermediate case between the Morse and the Floer settings.

Figures (14)

• Figure 1: A stabilized circle with an extra generator, but no index $2$ point to kill it.
• Figure 2: Relocating a Morse relation from an index $2$ critical point $x$ to an index $n$ critical point $z$.
• Figure 3: Augmentation like trajectories in the stable Morse setting, and their Floer analog.
• Figure 4: A bouncing trajectory in $\mathcal{M}^{\dag}(x_{-},x_{+})$, and its Floer analog. Hybrid trajectories are bouncing trajectories using different Morse data on the different components, which is depicted by a change of color.
• Figure 5: A stable Morse step through a critical point $y$.

Theorems & Definitions (49)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Definition 2.1
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Definition 4.1