[Paper Review] Morse homology on noncompact manifolds

Kai Cieliebak, Urs Frauenfelder|ArXiv.org|Nov 9, 2009

Geometric and Algebraic Topology5 references4 citations

TL;DR

This paper establishes the relationship between three variants of Morse homology on noncompact manifolds: Novikov-type homology $HM$, the direct-involution limit $ar{HM}$, and the inverse-direct limit $\underline{HM}$, under field coefficients. It proves that $ar{HM} \cong HM$ via an isomorphism, while the canonical map $\kappa: \bar{HM} \to \underline{HM}$ is surjective but not necessarily injective, resolving a key question in Floer theory and clarifying the algebraic structure of Morse homology in noncompact settings.

ABSTRACT

Given a Morse function on a manifold whose moduli spaces of gradient flow lines for each action window are compact up to breaking one gets a bidirect system of chain complexes. There are different possibilities to take limits of such a bidirect system. We discuss in this note the relation between these different limits.

Motivation & Objective

• To clarify the algebraic relationship between three distinct limits of Morse homology on noncompact manifolds: $HM$, $\bar{HM}$, and $\underline{HM}$.
• To determine under what conditions the canonical maps between these homology groups are isomorphisms, surjections, or injections.
• To resolve a question raised by K. Ono regarding whether the map $\bar{\rho}: HM \to \bar{HM}$ is an isomorphism in the context of Floer homology.
• To provide an algebraic explanation for why $HM \cong \bar{HM}$ in cases like weakly monotone symplectic manifolds, where direct computation previously showed equality.
• To demonstrate via counterexample that $\kappa: \bar{HM} \to \underline{HM}$ need not be injective, even when $\bar{HM} \neq 0$.

Proposed method

• The authors define Morse homology $HM^{[a,b]}_*$ for fixed action windows $[a,b]$, assuming compactness of moduli spaces of gradient flow lines up to breaking.
• They construct a bidirect system of chain complexes indexed by action windows $[a,b]$, enabling the formation of inverse and direct limits.
• They define three homology groups: $HM$ via Novikov completion of the chain complex, $\bar{HM} = \varinjlim_{b \to \infty} \varprojlim_{a \to -\infty} HM^b_a$, and $\underline{HM} = \varprojlim_{a \to -\infty} \varinjlim_{b \to \infty} HM^b_a$.
• They analyze the canonical maps $\kappa: \bar{HM} \to \underline{HM}$, $\bar{\rho}: HM \to \bar{HM}$, and $\underline{\rho}: HM \to \underline{HM}$, proving their properties under field coefficients.
• They use an explicit counterexample with a disjoint union of lines $\bigsqcup_{n=1}^\infty \mathbb{R}$ to show $\kappa$ is not injective, while $\bar{HM} \neq 0$.
• They prove $\bar{\rho}$ is an isomorphism and $\kappa$ is surjective by analyzing the structure of homology classes and using divisibility arguments in $\mathbb{Z}$-modules.

Experimental results

Research questions

• RQ1 Is the canonical map $\bar{\rho}: HM \to \bar{HM}$ an isomorphism when coefficients are a field?
• RQ2 What is the relationship between $\bar{HM}$ and $\underline{HM}$, and is the map $\kappa: \bar{HM} \to \underline{HM}$ injective?
• RQ3 Does the isomorphism $HM \cong \bar{HM}$ hold in general for noncompact manifolds with field coefficients?
• RQ4 Can the algebraic structure of Morse homology on noncompact manifolds be axiomatized to extend beyond finite-dimensional settings?
• RQ5 Why does $\kappa$ fail to be injective in general, and what structural features of the system cause this?

Key findings

• The map $\bar{\rho}: HM \to \bar{HM}$ is an isomorphism when homology is computed with field coefficients, establishing that the Novikov completion and the direct-involution limit agree.
• The map $\kappa: \bar{HM} \to \underline{HM}$ is surjective but not necessarily injective, as shown by a counterexample with $\underline{HM} = 0$ and $\bar{HM} \neq 0$.
• The counterexample involves a disjoint union of lines $\bigsqcup_{n=1}^\infty \mathbb{R}$ with Morse functions having critical points at $f(\overline{c}_n) = n$ and $f(\underline{c}_n) = -n$, leading to nontrivial $\bar{HM}$ but trivial $\underline{HM}$.
• The non-injectivity of $\kappa$ arises from the failure of Mittag-Leffler condition in the inverse limit system, particularly due to unbounded growth in transition maps.
• The proof that $\bar{HM} = 0$ in the counterexample relies on divisibility by arbitrarily high powers of 2 in the transition maps, forcing coefficients to vanish.
• The existence of a nontrivial class $\xi = \sum a_j \underline{c}_j$ in $HM$ with $a_j \in \{0,1\}$, not in the image of $\partial$, shows $HM \neq 0$, while $\bar{HM} = 0$, confirming the counterexample.