ncts

Abstract: In 2016, Tsai, Tseng, and Yau utilized the Lefchetz decomposition to define a family of

A_{\infty}

-algebras

F^{p} (M, ω)

for a symplectic manifold

(M^{2 n}, ω)

and

p = 0, 1, \dots, n - 1

. In 2018, Tanaka and Tseng showed that for each

p

, the

A_{\infty}

-algebra

F^{p} (M, ω)

is quasi-isomorphic to a mapping cone cdga associated to the chain map

ω^{p + 1} \land : Ω^{∙} (M) ⟶ Ω^{∙ + 2 p + 2} (M) .

In general, given a closed form

ψ

M

one may define a mapping cone complex

C (M, ψ)

, which, in addition, has a product structure when the degree

| ψ |

is even. Its cohomology

H_{C} (M, ψ)

is invariant under diffeomorphisms preserving the de Rham class

{[ψ]}_{d R}

. This cohomology possesses certain properties analogous to those of de Rham cohomology.

By proving mapping cone versions of the Leray--Serre spectral sequence, we establish a theory for computing the cone cohomology of fibrations equipped with compatible forms. In the symplectic context, this applies to symplectic products and symplectic fibrations.

Organizers: Nicolau Sarquis Aiex (NTNU), Siao-Hao Guo (NTU), Chao-Ming Lin (NTU), Wei-Bo Su (NCU), Chung-Jun Tsai (NTU)