On a spectral sequence for twisted cohomologies

TL;DR

This paper constructs a spectral sequence for twisted de Rham cohomology on compact manifolds, linking differentials to cup and Massey products, thus generalizing Atiyah and Segal's results.

Contribution

It introduces a spectral sequence converging to twisted de Rham cohomology and expresses its differentials via cup and Massey products, extending previous work.

Findings

Spectral sequence converges to twisted de Rham cohomology.

Differentials expressed through cup and Massey products.

Results on indeterminacy of differentials.

Abstract

Let ($\Omega^{\ast}(M), d$) be the de Rham cochain complex for a smooth compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$, there are a twisted de Rham cochain complex $(\Omega^{\ast}(M), d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the filtration $F_p(\Omega^{\ast}(M))=\bigoplus_{i\geq p}\Omega^i(M)$ of $\Omega^{\ast}(M)$, which converges to the twisted de Rham cohomology $H^*(M,H)$. We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.