AO with Structured Manifolds

• AO with Structured Manifolds is an exploration of A∞-operations constructed from sphere bundles over symplectic manifolds, linking topology and symplectic invariants.
• The methodology uses mapping cone constructions and filtered differential forms to establish quasi-isomorphisms between A∞-structures and de Rham algebras.
• The study reveals Calabi–Yau properties and intersection pairings, providing actionable insights on coisotropic and isotropic chain intersections in symplectic geometry.

In the context of symplectic geometry, "AO with Structured Manifolds" refers to the study of $A_\infty$-algebras (here denoted as "AO," an abbreviation for $A_\infty$-operations) arising from geometric data that is enhanced by nontrivial topological structures, specifically certain sphere bundles over symplectic manifolds. The construction and analysis of such $A_\infty$-algebras, particularly in the work of Tsai–Tseng–Yau and as topologically interpreted by Tanaka and Tseng, establish deep links between filtered cohomological invariants of symplectic manifolds and the algebraic structures on their associated sphere bundles. The resulting $A_\infty$-algebras, their structural maps, equivalences, and intersection-theoretic interpretations provide a refined framework for both homotopical algebra and intersection theory in symplectic topology (Tanaka et al., 2017).

1. Topological Foundations: Odd Sphere Bundles over Symplectic Manifolds

Let $M$ be a closed symplectic manifold with integral symplectic form $[\omega] \in H^2(M; \mathbb{Z})$. The corresponding prequantum line bundle $L \to M$ has first Chern class $c_1(L) = [\omega]$ with the unit circle bundle

$S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$

More generally, for each $0 \leq p \leq n = \tfrac12 \dim M$, the rank-$A_\infty$0 complex bundle $A_\infty$1 with Euler class $A_\infty$2 admits an associated odd sphere bundle

$A_\infty$3

with Euler class $A_\infty$4. The real cohomology of $A_\infty$5, via the Gysin sequence, satisfies

$A_\infty$6

where $A_\infty$7 denotes the $A_\infty$8-filtered cohomology associated with the symplectic form.

At the level of differential graded algebras (cdgas), it is shown that

$A_\infty$9

with explicit mapping cone structure: $A_\infty$0 This provides a topological model for the algebraic structures arising from $A_\infty$1 and its associated sphere bundles (Tanaka et al., 2017).

2. $A_\infty$2 Structures from Filtered Differential Forms

Tsai–Tseng–Yau introduced a sequence of $A_\infty$3-algebras derived from the complex $A_\infty$4 of $A_\infty$5-filtered differential forms, forming

$A_\infty$6

where $A_\infty$7 are symplectic first-order operators.

The $A_\infty$8-structure is characterized by nonvanishing products $A_\infty$9, specifically:

• $A_\infty$0 is the differential of degree $A_\infty$1
• $A_\infty$2 for $A_\infty$3, with correction terms (subtraction of $A_\infty$4 components) otherwise
• $A_\infty$5 appears in higher degrees, encoding further nontrivial algebraic data

All higher products vanish ($A_\infty$6 for $A_\infty$7), and the induced cohomology ring recovers $A_\infty$8. This $A_\infty$9-algebra is graded commutative at the product level and encodes both the geometric and symplectic data of $M$0 (Tanaka et al., 2017).

3. Quasi-Isomorphism to the De Rham Algebra on Sphere Bundles

The mapping cone cdga $M$1 associated with the sphere bundle $M$2 admits explicit chain maps to and from the $M$3-algebra $M$4: $M$5 with $M$6 and $M$7 for an explicit homotopy $M$8. This exhibits $M$9 as a strong deformation retract of $[\omega] \in H^2(M; \mathbb{Z})$0, yielding a quasi-isomorphism between their $[\omega] \in H^2(M; \mathbb{Z})$1-structures.

Furthermore, the $[\omega] \in H^2(M; \mathbb{Z})$2-map structure is given by setting $[\omega] \in H^2(M; \mathbb{Z})$3, $[\omega] \in H^2(M; \mathbb{Z})$4, $[\omega] \in H^2(M; \mathbb{Z})$5 $[\omega] \in H^2(M; \mathbb{Z})$6, compatible with the homotopy algebra relations. This equivalence demonstrates that $[\omega] \in H^2(M; \mathbb{Z})$7, solidifying the correspondence between the filtered algebraic invariants and the de Rham algebra on structured sphere bundles (Tanaka et al., 2017).

4. Calabi–Yau Property of the $[\omega] \in H^2(M; \mathbb{Z})$8-Algebras

An $[\omega] \in H^2(M; \mathbb{Z})$9-algebra $L \to M$0 is said to be Calabi–Yau of dimension $L \to M$1 if it admits a degree $L \to M$2 pairing $L \to M$3 that is nondegenerate on cohomology, graded symmetric, and cyclic: $L \to M$4 For $L \to M$5, the Poincaré duality pairing $L \to M$6 endows it with the Calabi–Yau structure of dimension $L \to M$7. This structure is transported to $L \to M$8 via the quasi-isomorphism, with the explicit formula

$L \to M$9

where $c_1(L) = [\omega]$0 is the reflection star operator. Nondegeneracy, symmetry, and cyclicity are verified using harmonic representatives and Hodge-theoretic considerations. The Calabi–Yau property is fundamental for the application of these algebras in intersection theory and for understanding their role as invariants of structured symplectic manifolds (Tanaka et al., 2017).

5. Intersection Theory for Coisotropic and Isotropic Chains

Specializing to $c_1(L) = [\omega]$1, the filtered forms $c_1(L) = [\omega]$2 are the primitive forms $c_1(L) = [\omega]$3, and the derived complex $c_1(L) = [\omega]$4 decomposes as

$c_1(L) = [\omega]$5

Utilizing de Rham duality:

• A primitive $c_1(L) = [\omega]$6-current $c_1(L) = [\omega]$7 is Poincaré dual to a coisotropic submanifold of codimension $c_1(L) = [\omega]$8.
• Its symplectic mirror $c_1(L) = [\omega]$9 is dual to an isotropic submanifold.

This framework produces a chain complex with coisotropic and isotropic chains: $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$0 where $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$1 denotes coisotropic chains of dimension $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$2, and $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$3 denotes isotropic chains of dimension $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$4.

The intersection pairing is realized via lifting coisotropic and isotropic cycles to submanifolds $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$5: $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$6 presenting the Calabi–Yau pairing as an intersection number in the associated sphere bundle. For concrete illustration, the Kodaira–Thurston manifold with symplectic form $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$7 features primitive cohomology generators dual to coisotropic and isotropic 2-tori whose intersection numbers correspond to their lifts in $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$8 (Tanaka et al., 2017).

6. Functoriality and Extension to Symplectic Maps and Correspondences

The assignment $S^1 \longrightarrow E_0 \xrightarrow{\pi} M.$9 is functorial for smooth maps $0 \leq p \leq n = \tfrac12 \dim M$0 satisfying $0 \leq p \leq n = \tfrac12 \dim M$1. This functoriality extends to the mapping cone constructions $0 \leq p \leq n = \tfrac12 \dim M$2, and the assignment becomes a homotopy sheaf on $0 \leq p \leq n = \tfrac12 \dim M$3. Moreover, these structures extend to bimodules over Lagrangian or isotropic correspondences, laying the analytic foundation for Weinstein functoriality in the smooth category. This functorial behavior supports the applicability of these $0 \leq p \leq n = \tfrac12 \dim M$4-invariants in broader contexts of symplectic topology and its categorical formulations (Tanaka et al., 2017).