Knowlpedia

Definition

de Rham homotopy operator

The degree-minus-one operator obtained by integrating a pulled-back form along a smooth homotopy, yielding a chain homotopy between endpoint pullbacks.

Definition

Let F:M×[0,1]→NF:M\times[0,1]\to N be a , with Ft(x)=F(x,t)F_t(x)=F(x,t). The de Rham homotopy operator associated to FF is the degree-−1-1

KF:Ωk(N)⟶Ωk−1(M),KFω=∫01ι∂t(F∗ω) dt.K_F:\Omega^k(N)\longrightarrow\Omega^{k-1}(M),\qquad K_F\omega=\int_0^1\iota_{\partial_t}(F^*\omega)\,dt.

Here contraction extracts the component of the containing dtdt, and integration removes the interval variable. With this sign and interval orientation, the operator satisfies

dKF+KFd=F1∗−F0∗.dK_F+K_Fd=F_1^*-F_0^*.

It is therefore a concrete between the two endpoint pullback maps.

Chain-homotopy interpretation

The endpoint pullbacks F0∗F_0^* and F1∗F_1^* are cochain maps between . The displayed identity says precisely that KFK_F is a cochain homotopy between them. It follows that smoothly homotopic maps induce the same homomorphism on de Rham cohomology Tu, Chapter 17.

The formula follows from Cartan's identity for the and the fundamental theorem of calculus applied in the interval direction. Reversing the orientation of [0,1][0,1] or defining contraction with the opposite product order changes the overall sign.

Canonical contraction example

On a star-shaped open set U⊆RnU\subseteq\mathbb R^n, the radial homotopy F(x,t)=txF(x,t)=tx joins the constant map at the origin to the identity. For a closed form ω\omega of positive degree,

ω=d(KFω),\omega=d(K_F\omega),

because pullback by the constant endpoint vanishes in positive degree. This is the standard homotopy-operator proof of the Bott–Tu, Chapter I.

Scope

The operator depends on the chosen homotopy, not only on its endpoints, but the induced equality on cohomology does not. For k=0k=0, the target Ω−1(M)\Omega^{-1}(M) is understood to be zero; the homotopy formula then says that endpoint pullbacks agree on locally constant functions. Boundary or support conditions require checking that fiber integration preserves the relevant class of forms.

References
  1. Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. Publisher record. Relevant: Chapter I, homotopy operators, the Poincaré lemma, and homotopy invariance.
  2. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. Publisher record. Relevant: Chapter 17, the homotopy operator.