PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-10

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De rham cohomology is a contravariant functor

Statement

De Rham cohomology is contravariant: for smooth $F : M \to N$ and $G : N \to P$ , $(G \circ F)^{*} = F^{*} \circ G^{*}$ , and ${id}_{M}^{*} = {id}_{H^{k} (M)}$ .

Facts & Assumptions

Given: Composable smooth maps $F : M \to N$ , $G : N \to P$ and an integer $k$ .

[F1]

Pullback induces a well defined map on de rham cohomology: For a smooth $F : M \to N$ , the formula $F^{*} [ω] = [F^{*} ω]$ defines a linear map $H_{d R}^{k} (N) \to H_{d R}^{k} (M)$ for every integer $k$ .

[F2]

Pullback of forms is smooth functorial and preserves wedges: For a smooth map $F : M \to N$ , pullback sends smooth differential forms on $N$ to smooth differential forms on $M$ , is functorial, and satisfies $F^{*} (α \land β) = F^{*} α \land F^{*} β .$

[F3]

Homology respects identities and composition: For every $n \in Z$ : 1. $H_{n} (1_{C}) = 1_{H_{n} (C)}$ for every chain complex $C$ . 2. If $f : C \to D$ and $g : D \to E$ are chain maps, then $H_{n} (g \circ f) = H_{n} (g) \circ H_{n} (f) .$

Proof

technique · direct

1.1

On forms, pullback satisfies $(G \circ F)^{*} = F^{*} G^{*}$ and ${id}^{*} = id$ . These are identities between cochain maps, with arrows from $P$ to $M$ .

F2given

2.1

Using $C_{n} = Ω^{- n}$ , apply homology at degree $- k$ to these identities. Homology preserves identities and composition; the induced maps are those already defined on de Rham classes. This gives both identities in the statement.

F1F3step 1.1∎

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 17, pp.441–443, Proposition 17.2 and Corollary 17.3; local quotient calculations below.

Depends on

Used by

The pullback on cohomology reverses composition order Counterexample
De rham cohomology is a covariant functor False statement
Pullback is a homomorphism of de rham cohomology algebras Proposition
De rham cohomology is smooth homotopy invariant Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)