[Paper Review] S^1-equivariant Morse cohomology

Marko Berghoff|arXiv (Cornell University)|Apr 12, 2012

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper constructs $S^1$-equivariant Morse cohomology for smooth closed $S^1$-manifolds of dimension less than 3 by deforming the standard Morse complex to include 'k-jump' flow lines that incorporate finite-time $S^1$-orbits. The key result establishes isomorphism between the resulting equivariant Morse cohomology and ordinary equivariant cohomology, providing a finite-dimensional alternative to infinite-dimensional homotopy quotient constructions.

ABSTRACT

We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.

Motivation & Objective

• To develop a finite-dimensional Morse-theoretic approach to $S^1$-equivariant cohomology, avoiding infinite-dimensional homotopy quotients.
• To generalize classical Morse theory by incorporating $S^1$-symmetry into the coboundary operator through 'k-jump' flow lines.
• To prove that the resulting $S^1$-equivariant Morse cohomology is isomorphic to standard equivariant cohomology for closed $S^1$-manifolds of dimension less than 3.
• To address foundational issues such as transversality, compactness, and gluing of moduli spaces of k-jump flow lines.
• To explore the feasibility of extending the construction to non-closed manifolds and other Lie groups, and to address orientability over $\mathbb{Z}$-coefficients.

Proposed method

• Introduces an equivariant Morse complex where the differential counts gradient flow lines that may 'jump' along $S^1$-orbits for finite time intervals, defined via $k$-jump flow lines.
• Defines moduli spaces $\tilde{\mathcal{M}}_k(x,y)$ of $k$-jump flow lines between critical points $x$ and $y$, and establishes transversality and compactness under generic perturbations.
• Constructs a gluing map for broken flow lines, proving that the composition of the differential with itself vanishes, i.e., $d^2 = 0$.
• Uses continuation maps and cochain homotopies to show invariance under homotopy of the perturbation data, ensuring quasi-isomorphism invariance.
• Applies the continuation principle to prove that the cohomology is independent of the choice of perturbation, and hence well-defined.
• Relies on the structure of $BS^1 = \mathbb{CP}^\infty$ and the fact that $S^1$-orbits have a unique orbit type, enabling the construction of finite-dimensional models.

Experimental results

Research questions

• RQ1 Can $S^1$-equivariant cohomology be computed using a finite-dimensional Morse-theoretic complex instead of infinite-dimensional homotopy quotients?
• RQ2 How can the standard Morse differential be modified to incorporate $S^1$-symmetry through finite-time orbit jumps?
• RQ3 Is the resulting $S^1$-equivariant Morse cohomology isomorphic to the standard equivariant cohomology for closed $S^1$-manifolds of dimension less than 3?
• RQ4 What are the necessary geometric and analytic conditions (transversality, compactness, gluing) for the moduli spaces of $k$-jump flow lines?
• RQ5 Can the construction be extended to non-closed manifolds or other Lie groups like $T^n$?

Key findings

• The $S^1$-equivariant Morse cohomology defined via $k$-jump flow lines is isomorphic to the standard equivariant cohomology $H^*_{S^1}(M)$ for smooth closed $S^1$-manifolds of dimension less than 3.
• The differential $d_{S^1}$ on the equivariant Morse complex satisfies $d_{S^1}^2 = 0$, ensuring a well-defined cohomology theory.
• Transversality and compactness of the moduli spaces $\tilde{\mathcal{M}}_k(x,y)$ are established under generic perturbations of the vector field.
• Gluing of broken flow lines is well-defined and compatible with the differential, ensuring the complex is a chain complex.
• Continuation maps between different perturbations are cochain homotopic, proving invariance of the cohomology under homotopy of the perturbation data.
• The construction relies crucially on the structure of $BS^1 = \mathbb{CP}^\infty$ and the unique orbit type of $S^1$-actions, which limits direct generalization to other Lie groups.