nLab Cartan's homotopy formula

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Statement

Let $M$ be a differentiable manifold, $X$ a vector field on $M$, and $\mathcal{L}_X$ the Lie derivative along $X$. Denote the contraction of a vector field and a differential form $\omega$ by $\iota(X)(\omega)$.

Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says

$\mathcal{L}_X \omega = d \iota(X)\omega + \iota(X) d\omega$

The word “homotopy” is used because it supplies a homotopy operator for some manipulation with chain complexes in de Rham cohomology. Cartan’s homotopy formula is part of Cartan calculus.

See also noncommutative differential calculus where the formula is incorporated into the notion of Batalin-Vilkovisky module over a Gerstenhaber algebra.

References

The original reference:

• Élie Cartan, Leçons sur les invariants intégraux (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).