# nLab twisted de Rham cohomology

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential

$d + H \wedge(-) : \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.

## Properties

Twisted de Rham cohomology is the recipient of the twisted Chern character in twisted differential K-theory.

Created on May 20, 2011 06:49:35 by Urs Schreiber (131.211.238.38)