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

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Pullback induces a well defined map on de rham cohomology

Statement

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 $k$ .

Facts & Assumptions

Given: A smooth $F$ and a closed $k$ -form $ω$ on $N$ .

[F1]

De rham cohomology: The real de Rham cohomology is $H_{d R}^{k} (M) = Z^{k} (M) / B^{k} (M)$ , with $Z^{k}, B^{k}$ as in def-closed-and-exact-differential-forms. This is def-cohomology-object-of-a-cochain-complex in real vector spaces. Only a closed form $ω$ represents a class $[ω]$ . For closed forms $ω, ω'$ , equality $[ω] = [ω']$ means precisely $ω' - ω = d η$ for some $(k - 1)$ -form $η$ . Addition and real scalar multiplication are induced by those of forms. All groups on the empty manifold are zero.

[F2]

Pullback is a morphism of de rham complexes: A smooth map $F : M \to N$ induces a degree-zero real cochain map $F^{*} : Ω^{∙} (N) \to Ω^{∙} (M)$ .

[F3]

A chain map induces a well-defined map on homology: Let $f : C_{∙} \to D_{∙}$ be a chain map. For every $n \in Z$ there is a unique morphism $H_{n} (f) : H_{n} (C) \to H_{n} (D)$ such that the quotient maps from cycles to homology commute with $Z_{n} (f)$ .

Proof

technique · direct

1.1

The cochain identity gives $d F^{*} ω = F^{*} d ω = 0$ . If $ω' = ω + d η$ , then $F^{*} ω' - F^{*} ω = F^{*} d η = d (F^{*} η)$ , so both representatives produce the same class.

F1F2given

2.1

Reindex by $C_{n} = Ω^{- n} (N)$ and $D_{n} = Ω^{- n} (M)$ , with unchanged differentials. The cochain identity is the chain-map identity; its cycles and boundaries at $- k$ are exactly $Z^{k}$ and $B^{k}$ . The induced-homology theorem therefore gives the displayed linear map, agreeing with step 1.1 by its quotient property.

F3step 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

Smoothly homotopic maps induce the same de rham map Corollary
De rham cohomology is a contravariant functor Proposition
Pullback is a homomorphism of de rham cohomology algebras Proposition

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)