Finite selection of relation generators in the stable Morse fundamental group presentation

Determine an intrinsic, effective procedure to select a finite subset of the potentially infinite family of relation generators used to define the normal subgroup R in the stable Morse fundamental group construction for generic stable Morse data Xi = (f, X) on a closed manifold M with base point ⋆, so that the resulting quotient L/R still presents π1(M,⋆). The goal is to specify a concrete selection criterion or algorithm, derived from the moduli spaces and “bouncing” trajectories described in the paper, that yields a finite generating set of relations sufficient to produce the isomorphism L/R ≅ π1(M,⋆).

Background

The paper constructs a group L of “Morse loops” and a normal subgroup R generated by preferred “Morse relations” from moduli spaces of augmentation-like and bouncing trajectories associated to generic stable Morse data Xi = (f, X) on M × R^{N_+} × R^{N_-}. The evaluation map then induces an isomorphism L/R ≅ π1(M,⋆).

Unlike in classical Morse theory, the stable setting may produce an infinite family of generators for R. While it is known a posteriori that only finitely many relations are needed to present π1(M,⋆), the authors’ construction does not presently provide a method to identify such a finite subset from the infinite family, motivating the need for a concrete selection procedure consistent with their dynamical framework.