References on de Rham and Poincaré via sheaves












1












$begingroup$


I'm trying to understand the proof of de Rham's theorem and Poincaré duality via sheaf cohomology, as in the appendix to Hubbard's book ("Teichmuller spaces and applications...") and Conlon's ("Differentiable manifolds") book. They prove it by constructing a double complex from the fine resolution
$$
0tomathbb{R}_MtoOmega_0toOmega_1to dots
$$
and deriving the isomorphism of cohomologies from corresponding double complex. I would like to "updgrade" these results and see that the isomorphism thus obtained is explicitly given by integration.



Similarly, in the proof of Poincaré duality (obtained by applying a similar argument to the sheaf of currents), I'm interested in checking that the pairing of $k$-th and $(n-k)$-th de Rham cohomology spaces is given by $int_M omega_1wedgeomega_2$.



There are hints in Hubbard's book that this is possible (and not very difficult), in the form of Exercise A9.5 and Exercise A9.7. On the other hand, the remark after Theorem D.3.11 in Conlon's book suggests that the sheaf-theoretic proof is not well suited for that.



Are there any references where this is written down?






algebraic-topology manifolds sheaf-cohomology







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I'm trying to understand the proof of de Rham's theorem and Poincaré duality via sheaf cohomology, as in the appendix to Hubbard's book ("Teichmuller spaces and applications...") and Conlon's ("Differentiable manifolds") book. They prove it by constructing a double complex from the fine resolution
    $$
    0tomathbb{R}_MtoOmega_0toOmega_1to dots
    $$
    and deriving the isomorphism of cohomologies from corresponding double complex. I would like to "updgrade" these results and see that the isomorphism thus obtained is explicitly given by integration.



    Similarly, in the proof of Poincaré duality (obtained by applying a similar argument to the sheaf of currents), I'm interested in checking that the pairing of $k$-th and $(n-k)$-th de Rham cohomology spaces is given by $int_M omega_1wedgeomega_2$.



    There are hints in Hubbard's book that this is possible (and not very difficult), in the form of Exercise A9.5 and Exercise A9.7. On the other hand, the remark after Theorem D.3.11 in Conlon's book suggests that the sheaf-theoretic proof is not well suited for that.



    Are there any references where this is written down?






    algebraic-topology manifolds sheaf-cohomology







    share|cite|improve this question









    $endgroup$















      1












      1








      1


      1



      $begingroup$


      I'm trying to understand the proof of de Rham's theorem and Poincaré duality via sheaf cohomology, as in the appendix to Hubbard's book ("Teichmuller spaces and applications...") and Conlon's ("Differentiable manifolds") book. They prove it by constructing a double complex from the fine resolution
      $$
      0tomathbb{R}_MtoOmega_0toOmega_1to dots
      $$
      and deriving the isomorphism of cohomologies from corresponding double complex. I would like to "updgrade" these results and see that the isomorphism thus obtained is explicitly given by integration.



      Similarly, in the proof of Poincaré duality (obtained by applying a similar argument to the sheaf of currents), I'm interested in checking that the pairing of $k$-th and $(n-k)$-th de Rham cohomology spaces is given by $int_M omega_1wedgeomega_2$.



      There are hints in Hubbard's book that this is possible (and not very difficult), in the form of Exercise A9.5 and Exercise A9.7. On the other hand, the remark after Theorem D.3.11 in Conlon's book suggests that the sheaf-theoretic proof is not well suited for that.



      Are there any references where this is written down?






      algebraic-topology manifolds sheaf-cohomology







      share|cite|improve this question









      $endgroup$




      I'm trying to understand the proof of de Rham's theorem and Poincaré duality via sheaf cohomology, as in the appendix to Hubbard's book ("Teichmuller spaces and applications...") and Conlon's ("Differentiable manifolds") book. They prove it by constructing a double complex from the fine resolution
      $$
      0tomathbb{R}_MtoOmega_0toOmega_1to dots
      $$
      and deriving the isomorphism of cohomologies from corresponding double complex. I would like to "updgrade" these results and see that the isomorphism thus obtained is explicitly given by integration.



      Similarly, in the proof of Poincaré duality (obtained by applying a similar argument to the sheaf of currents), I'm interested in checking that the pairing of $k$-th and $(n-k)$-th de Rham cohomology spaces is given by $int_M omega_1wedgeomega_2$.



      There are hints in Hubbard's book that this is possible (and not very difficult), in the form of Exercise A9.5 and Exercise A9.7. On the other hand, the remark after Theorem D.3.11 in Conlon's book suggests that the sheaf-theoretic proof is not well suited for that.



      Are there any references where this is written down?






      algebraic-topology manifolds sheaf-cohomology




      algebraic-topology manifolds sheaf-cohomology






      share|cite|improve this question













      share|cite|improve this question











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      asked Jan 11 at 14:24









      Kostya_IKostya_I

      508213




      508213






















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          I think I have figured it out on my own. Let $pi:Omega^k(U)to C^k(U)$ denote the map that associates to a differential form the singular cochain given by the integration. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{dR}$, one constructs a sequence of cochains $$x_0in C^k(mathfrak{U},Omega^0), ; y_0in C^{k-1}(mathfrak{U},Omega^0),; x_1in C^{k-1}(mathfrak{U},Omega^1),dots,x_kin C^{0}(mathfrak{U},Omega^1),y_kin Omega_k$$ such that $x_i=d_text{dR}y_{i-1}=d_text{sh}x_i$. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{sing}$, one constructs a sequence of cochains $hat{x}_i,hat{y}_i$ with the same properties, but $d_text{dR}$ replaced by singular coboundary operator. All one has to check is that one can choose $hat{x}_i=pi(x_i)$ and $hat{y}_i=pi(y_i)$. This sequence works since $pi$ commutes with differentials in both directions. Similar arguments apply to the proof of Poincaré duality.






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            I think I have figured it out on my own. Let $pi:Omega^k(U)to C^k(U)$ denote the map that associates to a differential form the singular cochain given by the integration. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{dR}$, one constructs a sequence of cochains $$x_0in C^k(mathfrak{U},Omega^0), ; y_0in C^{k-1}(mathfrak{U},Omega^0),; x_1in C^{k-1}(mathfrak{U},Omega^1),dots,x_kin C^{0}(mathfrak{U},Omega^1),y_kin Omega_k$$ such that $x_i=d_text{dR}y_{i-1}=d_text{sh}x_i$. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{sing}$, one constructs a sequence of cochains $hat{x}_i,hat{y}_i$ with the same properties, but $d_text{dR}$ replaced by singular coboundary operator. All one has to check is that one can choose $hat{x}_i=pi(x_i)$ and $hat{y}_i=pi(y_i)$. This sequence works since $pi$ commutes with differentials in both directions. Similar arguments apply to the proof of Poincaré duality.






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              $begingroup$

              I think I have figured it out on my own. Let $pi:Omega^k(U)to C^k(U)$ denote the map that associates to a differential form the singular cochain given by the integration. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{dR}$, one constructs a sequence of cochains $$x_0in C^k(mathfrak{U},Omega^0), ; y_0in C^{k-1}(mathfrak{U},Omega^0),; x_1in C^{k-1}(mathfrak{U},Omega^1),dots,x_kin C^{0}(mathfrak{U},Omega^1),y_kin Omega_k$$ such that $x_i=d_text{dR}y_{i-1}=d_text{sh}x_i$. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{sing}$, one constructs a sequence of cochains $hat{x}_i,hat{y}_i$ with the same properties, but $d_text{dR}$ replaced by singular coboundary operator. All one has to check is that one can choose $hat{x}_i=pi(x_i)$ and $hat{y}_i=pi(y_i)$. This sequence works since $pi$ commutes with differentials in both directions. Similar arguments apply to the proof of Poincaré duality.






              share|cite|improve this answer









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                0












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                0





                $begingroup$

                I think I have figured it out on my own. Let $pi:Omega^k(U)to C^k(U)$ denote the map that associates to a differential form the singular cochain given by the integration. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{dR}$, one constructs a sequence of cochains $$x_0in C^k(mathfrak{U},Omega^0), ; y_0in C^{k-1}(mathfrak{U},Omega^0),; x_1in C^{k-1}(mathfrak{U},Omega^1),dots,x_kin C^{0}(mathfrak{U},Omega^1),y_kin Omega_k$$ such that $x_i=d_text{dR}y_{i-1}=d_text{sh}x_i$. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{sing}$, one constructs a sequence of cochains $hat{x}_i,hat{y}_i$ with the same properties, but $d_text{dR}$ replaced by singular coboundary operator. All one has to check is that one can choose $hat{x}_i=pi(x_i)$ and $hat{y}_i=pi(y_i)$. This sequence works since $pi$ commutes with differentials in both directions. Similar arguments apply to the proof of Poincaré duality.






                share|cite|improve this answer









                $endgroup$



                I think I have figured it out on my own. Let $pi:Omega^k(U)to C^k(U)$ denote the map that associates to a differential form the singular cochain given by the integration. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{dR}$, one constructs a sequence of cochains $$x_0in C^k(mathfrak{U},Omega^0), ; y_0in C^{k-1}(mathfrak{U},Omega^0),; x_1in C^{k-1}(mathfrak{U},Omega^1),dots,x_kin C^{0}(mathfrak{U},Omega^1),y_kin Omega_k$$ such that $x_i=d_text{dR}y_{i-1}=d_text{sh}x_i$. In the proof that $H^k(mathfrak{U},mathbb{R}_M)$ is isomorphic to $H^k_text{sing}$, one constructs a sequence of cochains $hat{x}_i,hat{y}_i$ with the same properties, but $d_text{dR}$ replaced by singular coboundary operator. All one has to check is that one can choose $hat{x}_i=pi(x_i)$ and $hat{y}_i=pi(y_i)$. This sequence works since $pi$ commutes with differentials in both directions. Similar arguments apply to the proof of Poincaré duality.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 12 at 15:46









                Kostya_IKostya_I

                508213




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