Simple Homotopy in Fukaya Categories

• Simple Homotopy Theory in Fukaya Categories is a framework that refines Floer theory by integrating π₁-data to capture Whitehead torsion and simple homotopy types of Lagrangians.
• It employs π₁-enhanced Floer complexes and graded, unobstructed Floer theory under c₁(X)=0 to provide strict control over mapping cone torsion and detect simple homotopy equivalences.
• This approach bridges algebraic invariants with topological insights, offering practical techniques to distinguish Lagrangian submanifolds beyond classical homotopy equivalence.

Simple homotopy theory in Fukaya categories refers to the program of refining the algebraic and categorical structures underlying the Floer theory of Lagrangian submanifolds, so as to recover not only classical homotopy equivalence but the finer, Whitehead-type invariants (such as Whitehead torsion and simple homotopy equivalence) by appropriately enriching the Fukaya category with fundamental group data. This refinement, realized by a π₁-linear enhancement of the Floer complexes and strict control of torsion in mapping cones of isomorphisms, enables the extraction of the simple homotopy type of Lagrangians and their relations within the symplectic manifold. The approach critically relies on the vanishing of the first Chern class (to ensure unobstructedness and full gradings), module structures over group rings ℤ[π₁(X)], and conditions on Isomorphisms—ultimately relating algebraic isomorphisms in the Fukaya category to simple homotopy equivalences.

1. π₁-Enhanced Floer Complexes

For exact Lagrangian submanifolds $K, L \subset (X, \omega)$, one considers the classical (wrapped or compact) Floer complex: $$CF^*(K, L) = \bigoplus_{x \in K \cap L} \mathbb{Z} \langle \mathfrak{o}_x \rangle$$ where $\mathfrak{o}_x$ are orientation lines. To integrate π₁(X)-data, choose lifts $\tilde{x} \in \widetilde{X}$ for each intersection, yielding the π₁-linear chain complex: $$\underline{CF}^*(K, L) = \bigoplus_{x \in K \cap L} \mathbb{Z} \langle \tilde{x} \rangle$$ The differential is augmented: for each pseudoholomorphic strip $u$ mapping asymptotically from $x$ to $y$, the lifted curve in the universal cover identifies an element $g(u) \in \pi_1(X)$ such that

$$\partial(\tilde{x}) = \sum_{u \in \mathcal{M}(y; x)} g(u) \cdot \tilde{y}$$

This structure makes $\underline{CF}^*(K, L)$ a (free, based) module over the group ring $\mathbb{Z}[\pi_1(X)]$.

2. Whitehead Torsion in Fukaya Categories

Given a based chain complex $C^*$ over $\mathbb{Z}[\pi_1(X)]$, Whitehead torsion $\tau(C^*) \in Wh(\pi_1(X))$ is defined classically via comparison of explicit bases with preferred acyclic splittings. The mapping cone of a morphism $\alpha \in CF^0(\mathcal{K}, \mathcal{L})$ (between twisted complexes derived from Lagrangians) is formed as: $$\text{Cone}(\alpha) = \begin{pmatrix} \mathcal{K}[1] \ \mathcal{L} \end{pmatrix}$$ with the associated differential. A simple isomorphism is one for which the mapping cone is simply acyclic, i.e., acyclic and with $\tau(\text{Cone}(\alpha)) = 1 \in Wh(\pi_1(X))$. Thus, in the Fukaya category Tw₍Ch₎𝓕(X), the classification of isomorphisms refines to a test of simplicity via (twisted) Whitehead torsion.

Notably, when the Fukaya category is generated by simply connected objects (e.g., Lefschetz thimbles in a Weinstein manifold), all isomorphisms are simple, since the group ring structure becomes trivial.

3. Vanishing of the First Chern Class and ℤ-Gradings

The hypothesis $c_1(X) = 0$ is central. This implies:

• Availability of a full $\mathbb{Z}$-grading for all Lagrangian branes.
• Decomposition of Floer complexes into graded pieces, facilitating the definition and preservation of torsion.
• Absence of obstructive bubbling, yielding unobstructedness and well-defined Floer theory (for both compact and wrapped settings) over $\mathbb{Z}$.

The lift of grading and coherent orientation data ensures that all constructions—differentials, higher A∞ operations, and mapping cones—are controlled at the chain level and compatible with torsion computations.

4. Simple Homotopy Equivalence for Closed Lagrangians

Suppose $X$ is a Weinstein manifold with $c_1(X) = 0$, and let $L \subset X$ be a closed exact Lagrangian such that $L \xrightarrow{\simeq} X$ is a homotopy equivalence. Let $K \subset X$ be another closed exact Lagrangian with the induced map $\pi_1(K) \rightarrow \pi_1(X)$ an isomorphism, and suppose $K$ is isomorphic to $L$ in the Fukaya category.

Then, the isomorphism gives a chain map between their cellular chain complexes (via the π₁-linear Floer complexes), and the construction yields: $$\tau(\operatorname{cone}(\alpha)) = 1 \quad \text{in } Wh\left(\pi_1(X)\right)$$ It follows that $K \rightarrow X$ is also a homotopy equivalence, and the composition $K \rightarrow X \rightarrow L$ is a simple homotopy equivalence. Thus, $K$ and $L$ have identical simple homotopy types; this is detected directly in the categorical framework via the vanishing Whitehead torsion of the Floer-theoretic mapping cone.

This conclusion is contingent upon the isomorphism at the level of $\pi_1$, ensuring that the module structures and resulting torsion invariants are faithful to the global topology of $X$.

5. Applications and Implications

The categorical refinement aligns the algebraic data in the Fukaya category with simple homotopy-theoretic invariants from algebraic topology, such as Whitehead and Reidemeister torsion. Consequently:

• Isomorphism classes in the Fukaya category of Weinstein manifolds with $c_1(X) = 0$ and trivial $\pi_1$-defect coincide with simple homotopy types of the underlying Lagrangians.
• The invariants can be computed concretely from the π₁-enhanced Floer complexes, yielding explicit comparison with cellular chain complexes.
• The approach yields new methods for detecting differences between Lagrangians with the same classical homotopy type but distinct simple homotopy types—a phenomenon not accessible to ordinary Floer or quantum invariants.
• The technique is particularly well-suited to distinguishing cotangent bundles of lens spaces and analyzing Dehn twists and their effect at the level of simple homotopy type.

6. Representative Formulæ

The key constructions and invariants are given explicitly by:

• The π₁-enhanced Floer differential:

$$\partial (\tilde{x}) = \sum_{u\in \mathcal{M}(y,x)} g(u)\,\tilde{y}$$

• The mapping cone torsion criterion:

$$\tau\left(\operatorname{cone}(\alpha)\right) = 1 \quad \text{in } Wh\left(\pi_1(X)\right)$$

• The comparison between the Floer and cellular complexes:

$$\underline{CF^*(K, K)} \simeq \underline{C^*_{\text{cell}}(K)}, \quad \underline{CF^*(L, L)} \simeq \underline{C^*_{\text{cell}}(L)}$$

In summary, the simple homotopy theory framework for Fukaya categories demonstrates that, under strong topological constraints ($c_1(X) = 0$ and π₁-isomorphisms), categorical isomorphisms correspond to simple homotopy equivalences, and the π₁-linear structure, together with Whitehead torsion, provides a powerful invariant to distinguish and identify Lagrangians at the refined simple-homotopical level (Kim, 6 Sep 2025).