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## at.algebraic topology – The judgement of a “relatively” flat connection

Suppose that $X$ is a related gentle manifold and $Gamma$ is a bunch appearing easily, freely, correctly and discretely on $X$, in order that $Y=X/Gamma$ is one other gentle manifold endowed with a masking map $pi:Xrightarrow Y$.

Suppose that $G$ is a equivocate group and that $rho:Gamma rightarrow G$ is a bunch homomorphism. Then we are able to assume the quotient $E_rho =(Xtimes G)/Gamma$, the place the motion of $Gamma$ on $G$ is given by composing $rho$ with the adjoint motion of $G$ on itself. $E_rho$ is of course a principal $G$-bundle over $Y$.

My query is that if there exists a situation for a principal $G$-bundle $E$ on $Y$ to breathe isomorphic to an $E_rho$, for some homomorphism $rho:Gamma rightarrow G$.

This can breathe interpreted as a “relative” judgement of a flat connection since, if $X$ is the common masking area of $Y$ and $Gamma=pi_1(Y)$, then the situation for $E$ to breathe of the design $E_rho$ is that $E$ admits a flat connection.

moreover, the similar query can breathe prolonged to the holomorphic class. For occasion, we are able to maintain $X$ a Riemann floor and $G=U(n)$. In that standing, if $X$ is the hyperbolic aircraft and $Y$ is a compact Riemann floor of genus $geq 2$, the situation for a holomorphic Hermitian vector bundle $E$ to breathe of the design $E_rho$ is that it’s steady of diploma $0$ (that is the Theorem of Narasimhan-Seshadri).

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