TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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De rham cohomology is smooth homotopy invariant

Statement

A smooth homotopy equivalence induces an isomorphism of de Rham graded real algebras.

Facts & Assumptions

Given: Smooth maps $f : M \to N$ , $g : N \to M$ with smooth homotopies $g f ≃ {id}_{M}$ and $f g ≃ {id}_{N}$ .

[F1]

Smoothly homotopic maps induce the same de rham map: Smoothly homotopic smooth maps induce equal maps on de Rham cohomology in every degree.

[F2]

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

[F3]

Pullback is a homomorphism of de rham cohomology algebras: Smooth pullback induces a unital graded real algebra homomorphism $H_{d R}^{∙} (N) \to H_{d R}^{∙} (M)$ .

Proof

technique · direct

1.1

By smooth-homotopy invariance and functoriality, $f^{*} g^{*} = (g f)^{*} = ({id}_{M})^{*} = {id}_{H^{∙} (M)}$ .

F1F2given

2.1

The other homotopy gives $g^{*} f^{*} = (f g)^{*} = {id}_{H^{∙} (N)}$ . Both maps preserve multiplication and units, so these two identities exhibit inverse graded algebra homomorphisms. The conclusion includes empty manifolds, since a homotopy equivalence to an empty manifold forces both to be empty.

F1F2F3step 1.1given∎

Source locator

Lee, Theorem 17.11, pp.445–446; this item assumes smooth homotopies explicitly, so does not use a continuous smoothing theorem.

Depends on

Used by

De rham cohomology is continuous homotopy invariant on smooth manifolds Corollary
De rham cohomology of punctured euclidean space Corollary
Homotopy equivalent annulus and circle have isomorphic de rham rings Example
De rham cohomology of spheres Theorem

Dependency tree · two levels

12 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)