nLab
twisted de Rham cohomology

Context

Differential cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For XX a smooth manifold and HΩ 3(X)H \in \Omega^3(X) a closed differential 3-form, the HH twisted de Rham complex is the 2\mathbb{Z}_2-graded vector space Ω even(X)Ω odd(X)\Omega^{even}(X) \oplus \Omega^{odd}(X) equipped with the HH-twisted de Rham differential

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

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

There is also the cohomology of the chain complex whose maps are just multiplication by HH.

H():Ω even/odd(X)Ω odd/even(X), H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,

This is also called H-cohomology (Cavalcanti 03, p. 19).

Properties

Proposition

If the de Rham complex (Ω (X),d dr)(\Omega^\bullet(X),d_{dr}) is formal, then for HΩ cl 3(X)H \in \Omega_{cl}^3(X) a closed differential 3-form, the HH-twisted de Rham cohomology of XX coincides with its H-cohomology, for any closed 3-form HH.

(Cavalcanti 03, theorem 1.6).

References

Revised on February 14, 2018 11:13:38 by Urs Schreiber (195.229.110.1)