# de Rham cohomology vs. iterated tangent bundles?

I have two related questions. Here $M$ is a real smooth manifold, $TM$ is its tangent bundle, $T^n M := T ... TM$ is the $n$-th iterated tangent bundle.

1. Fiberwise linear smooth functions $TM \to \mathbf R$ are the same as smooth one-forms on $M$. Is there a handy generalization of this to $n$-forms and some functions $T^n M \to \mathbf R$?

2. Can de Rham cohomology be expressed in terms of (the simplicial abelian group of) functions $T^\bullet M \to \mathbf R$?

Thank you.

• (Re: 1) differential forms on $M$ are functions on the supermanifold $\Pi TM$... May 24, 2011 at 15:21
• @Michael: sure! May 25, 2011 at 10:48

EDIT: I think an answer to your first question is explained in the papers:

• P.-A. Meyer, Qu'est ce qu'une différentielle d'ordre $$n$$, Exposition. Math. 7 (1989), 249–264.
• Laksov, Dan; Thorup, Anders, These are the differentials of order $$n$$. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353. Freely available online.

Quote from the second:

The higher order differentials were part of the folklore of mathematics up to the end of the previous century, and formulas like $$d^2f=f'_xd^2x+f'_yd^2y+{f''_{x^2}}dx^2+2{f''_{xy}}dxdy+{f''_{y^2}}dy^2$$ can be found in most classical calculus books. (...) the higher order differentials vanished (...) because the extensive user of exterior differentials led mathematicians to believe that $$d^2$$ should always be zero.

(...)

Let $$C_n:=\mathcal{C}^\infty(T^nX)$$. The differential of $$\mathcal{C}^\infty$$- functions on $$T^nX$$ can be viewed as a k-linear map $$d:C_n\rightarrow C_{n+1}$$, and we obtain a sequence of linear maps...

Then $$\Omega^n$$ is defined as a suitable submodule of $$C_n$$, and there is a product $$\Omega^p \otimes \Omega^n \rightarrow \Omega^{p+n}$$ such that $$d(\omega\cdot\pi)=d\omega\cdot\pi + \omega\cdot d\pi$$.

• Could you tell me how to get the paper of Meyer 1989? Thank you. Jul 3, 2021 at 0:22