Mixing
Mayer-Vietoris for twisted De Rham cohomology in Bott-Tu Differential Forms in Algebraic Topology
$begingroup$
Consider $E$ rank $k$ flat vector bundle over $M$ with locally constant trivialization $U_i$. Define $Omega^p(M,E)$ as compatible local sections $Omega^p(M,E)=Omega^p(M)otimes_{C^infty(M)} E$. Assume $E$ has two charts $Utimes R^k$ and $Vtimes R^k$ to obtain locally constant trivialization. Then consider the following sequence.
$0toOmega^star(M,E)toOmega^star(U,E)oplusOmega^star(V,E)toOmega^star(Ucap V,E)to 0$
The first map is just standard restriction and trivially injection. The second map is given by $(alpha,beta)to(phi_{UV}alpha-beta)$ where constant matrix $phi_{UV}$ is the corresponding linear map between $R^k$ factors of $Ucap Vtimes R^k$ and $Vcap Utimes R^k$ of $E$.
Consider partition of unity $rho_U,rho_V$ subordinate to covering $U,V$. Let $omegainOmega^star(Ucap V,E)$. Consider $(rho_V phi_{UV}^{-1}omega,-rho_Uomega)$. Note that $g_{UV}$ are constants acting only on $E$'s vector components. So $phi_{UV}rho_V phi_{UV}^{-1}omega+rho_Uomega=rho_Vomega+rho_Uomega=omega$. Hence, the sequence is exact.
$textbf{Q:}$ It seems that I can induce Mayer-Vietoris over forms with values in flat vector bundle. Is this correct?
Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 7.
general-topology geometry differential-geometry algebraic-topology differential-topology
asked Jan 23 at 21:04
user45765user45765
2,6702724
$endgroup$
add a comment |
$begingroup$
Consider $E$ rank $k$ flat vector bundle over $M$ with locally constant trivialization $U_i$. Define $Omega^p(M,E)$ as compatible local sections $Omega^p(M,E)=Omega^p(M)otimes_{C^infty(M)} E$. Assume $E$ has two charts $Utimes R^k$ and $Vtimes R^k$ to obtain locally constant trivialization. Then consider the following sequence.
$0toOmega^star(M,E)toOmega^star(U,E)oplusOmega^star(V,E)toOmega^star(Ucap V,E)to 0$
The first map is just standard restriction and trivially injection. The second map is given by $(alpha,beta)to(phi_{UV}alpha-beta)$ where constant matrix $phi_{UV}$ is the corresponding linear map between $R^k$ factors of $Ucap Vtimes R^k$ and $Vcap Utimes R^k$ of $E$.
Consider partition of unity $rho_U,rho_V$ subordinate to covering $U,V$. Let $omegainOmega^star(Ucap V,E)$. Consider $(rho_V phi_{UV}^{-1}omega,-rho_Uomega)$. Note that $g_{UV}$ are constants acting only on $E$'s vector components. So $phi_{UV}rho_V phi_{UV}^{-1}omega+rho_Uomega=rho_Vomega+rho_Uomega=omega$. Hence, the sequence is exact.
$textbf{Q:}$ It seems that I can induce Mayer-Vietoris over forms with values in flat vector bundle. Is this correct?
Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 7.
general-topology geometry differential-geometry algebraic-topology differential-topology
asked Jan 23 at 21:04
user45765user45765
2,6702724
$endgroup$
add a comment |
$begingroup$
Consider $E$ rank $k$ flat vector bundle over $M$ with locally constant trivialization $U_i$. Define $Omega^p(M,E)$ as compatible local sections $Omega^p(M,E)=Omega^p(M)otimes_{C^infty(M)} E$. Assume $E$ has two charts $Utimes R^k$ and $Vtimes R^k$ to obtain locally constant trivialization. Then consider the following sequence.
$0toOmega^star(M,E)toOmega^star(U,E)oplusOmega^star(V,E)toOmega^star(Ucap V,E)to 0$
The first map is just standard restriction and trivially injection. The second map is given by $(alpha,beta)to(phi_{UV}alpha-beta)$ where constant matrix $phi_{UV}$ is the corresponding linear map between $R^k$ factors of $Ucap Vtimes R^k$ and $Vcap Utimes R^k$ of $E$.
Consider partition of unity $rho_U,rho_V$ subordinate to covering $U,V$. Let $omegainOmega^star(Ucap V,E)$. Consider $(rho_V phi_{UV}^{-1}omega,-rho_Uomega)$. Note that $g_{UV}$ are constants acting only on $E$'s vector components. So $phi_{UV}rho_V phi_{UV}^{-1}omega+rho_Uomega=rho_Vomega+rho_Uomega=omega$. Hence, the sequence is exact.
$textbf{Q:}$ It seems that I can induce Mayer-Vietoris over forms with values in flat vector bundle. Is this correct?
Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 7.
general-topology geometry differential-geometry algebraic-topology differential-topology
asked Jan 23 at 21:04
user45765user45765
2,6702724
$endgroup$
Consider $E$ rank $k$ flat vector bundle over $M$ with locally constant trivialization $U_i$. Define $Omega^p(M,E)$ as compatible local sections $Omega^p(M,E)=Omega^p(M)otimes_{C^infty(M)} E$. Assume $E$ has two charts $Utimes R^k$ and $Vtimes R^k$ to obtain locally constant trivialization. Then consider the following sequence.
$0toOmega^star(M,E)toOmega^star(U,E)oplusOmega^star(V,E)toOmega^star(Ucap V,E)to 0$
The first map is just standard restriction and trivially injection. The second map is given by $(alpha,beta)to(phi_{UV}alpha-beta)$ where constant matrix $phi_{UV}$ is the corresponding linear map between $R^k$ factors of $Ucap Vtimes R^k$ and $Vcap Utimes R^k$ of $E$.
Consider partition of unity $rho_U,rho_V$ subordinate to covering $U,V$. Let $omegainOmega^star(Ucap V,E)$. Consider $(rho_V phi_{UV}^{-1}omega,-rho_Uomega)$. Note that $g_{UV}$ are constants acting only on $E$'s vector components. So $phi_{UV}rho_V phi_{UV}^{-1}omega+rho_Uomega=rho_Vomega+rho_Uomega=omega$. Hence, the sequence is exact.
$textbf{Q:}$ It seems that I can induce Mayer-Vietoris over forms with values in flat vector bundle. Is this correct?
Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 7.
general-topology geometry differential-geometry algebraic-topology differential-topology
general-topology geometry differential-geometry algebraic-topology differential-topology
asked Jan 23 at 21:04
user45765user45765
2,6702724
asked Jan 23 at 21:04
user45765user45765
2,6702724
asked Jan 23 at 21:04
user45765user45765
2,6702724
asked Jan 23 at 21:04
user45765user45765
2,6702724
asked Jan 23 at 21:04
user45765user45765
2,6702724
2,6702724
add a comment |
add a comment |
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