nLab
Poincaré lemma

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cohomology

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Differential geometry

synthetic differential geometry

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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

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cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form ω\omega which is closed, d dRω=0d_{dR}\omega = 0, is locally exact, ω| U=d dRκ\omega|_U = d_{dR}\kappa.”

          In more detail: if XX is contractible then for every closed differential form ωΩ cl k(X)\omega \in \Omega^k_{cl}(X) with k1k \geq 1 there exists a differential form λΩ k1(X)\lambda \in \Omega^{k-1}(X) such that

          ω=d dRλ. \omega = d_{dR} \lambda \,.

          Moreover, for ω\omega a smooth family of closed forms, there is a smooth family of λ\lambdas satisfying this condition.

          This statement has several more abstract incarnations. One is that it says that on a Cartesian space (or a complex polydisc) the de Rham cohomology (the holomorphic de Rham cohomology) vanishes in positive degree.

          Still more abstractly this says that the canonical morphisms of sheaves of chain complexes

          Ω dR \mathbb{R} \to \Omega^\bullet_{dR}
          Ω hol \mathbb{C} \to \Omega^\bullet_{hol}

          from the locally constant sheaf on the real numbers (the complex numbers) to the de Rham complex (holomorphic de Rham complex) is a stalk-wise quasi-isomorphism – hence an equivalence in the derived category and hence induce an equivalence in hyper-abelian sheaf cohomology. (The latter statement fails in general in complex algebraic geometry, see (Illusie 12, 1.) and see also at GAGA.) (A variant of such resolutions of constant sheaves for the case over Klein geometries are BGG resolutions.)

          The Poincaré lemma is a special case of the more general statement that the pullbacks of differential forms along homotopic smooth function are related by a chain homotopy.

          Statement

          Theorem

          Let f 1,f 2:XYf_1, f_2 : X \to Y be two smooth functions between smooth manifolds and Ψ:[0,1]×XY\Psi : [0,1] \times X \to Y a (smooth) homotopy between them.

          Then there is a chain homotopy between the induced morphisms

          f 1 *,f 2 *:Ω (Y)Ω (X) f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)

          on the de Rham complexes of XX and YY.

          In particular, the action on de Rham cohomology of f 1 *f_1^* and f 2 *f_2^* coincide,

          H dR (f 1 *)H dR (f 2 *). H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.

          Moreover, an explicit formula for the chain homotopy ψ:f 1f 2\psi : f_1 \Rightarrow f_2 is given by the “homotopy operator

          ψ:ω(x [0,1]ι t(Ψ t *ω)(x))dt. \psi : \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi_t^*\omega)(x) ) d t \,.

          Here ι t\iota_{\partial t} denotes contraction (see Cartan calculus) with the canonical vector field tangent to [0,1][0,1], and the integration is that of functions with values in the vector space of differential forms.

          Proof

          We compute

          d Yψ(ω)+ψ(d Xω) = [0,1]d Yι tΨ t *(ω)dt+ [0,1]ι tΨ t *(d Xω)dt = [0,1][d Y,ι t]Ψ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1]d [0,1]Ψ t *(ω) =Ψ 1 *ωΨ 0 *ω =f 2 *ωf 1 *ω, \begin{aligned} d_{Y} \psi(\omega) + \psi( d_X \omega ) & = \int_{[0,1]} d_Y \iota_{\partial_t} \Psi_t^*(\omega) d t + \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(d_X \omega) d t \\ & = \int_{[0,1]} [d_Y,\iota_{\partial_t}] \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \mathcal{L}_{t} \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \partial_t \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} d_{[0,1]} \Psi_t^* (\omega) \\ & = \Psi_1^* \omega - \Psi_0^* \omega \\ & = f_2^* \omega - f_1^* \omega \end{aligned} \,,

          where in the integral we used first that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula [d,ι t]= t[d,\iota_{\partial t}] = \mathcal{L}_t for the Lie derivative along the cylinder on XX and finally the Stokes theorem.

          The Poincaré lemma proper is the special case of this statement for the case that f 1=const yf_1 = const_y is a function constant on a point yYy \in Y:

          Corollary

          If a smooth manifold XX admits a smooth contraction

          X (id,0) id X×[0,1] Ψ X (id,1) const x X \array{ X \\ \downarrow^{\mathrlap{(id,0)}} & \searrow^{\mathrlap{id}} \\ X \times [0,1] & \stackrel{\Psi}{\to} & X \\ \uparrow^{\mathrlap{(id,1)}} & \nearrow_{\mathrlap{const_x}} \\ X }

          then the de Rham cohomology of XX is concentrated on the ground field in degree 0. Moreover, for ω\omega any closed form on XX in positive degree an explicit formula for a form λ\lambda with dλ=ωd \lambda = \omega is given by

          λ= [0,1]ι tΨ t *(ω)dt. \lambda = \int_{[0,1]} \iota_{\partial_t}\Psi_t^*(\omega) d t \,.
          Proof

          In the general situation discussed above we now have f 1 *=0f_1^* = 0 in positive degree.

          References

          A nice account collecting all the necessary background (in differential geometry) is in

          • Daniel Litt, The Poincaré lemma and de Rham cohomology (pdf)

          Discussion in complex analytic geometry is in

          • Luc Illusie, Around the Poincaré lemma, after Beilinson, talk notes 2012 (pdf)

          following

          • Alexander Beilinson, pp-adic periods and de Rham cohomology, J. of the AMS 25 (2012), 715-738

          Last revised on October 2, 2017 at 16:56:38. See the history of this page for a list of all contributions to it.