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I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where $\mathcal G=G\times X$ where $G$ is a linear algebraic group (affine) and $X$ is the curve. Using the Kenneth formula, it suffishs to calculate both $\pi_1(X)$ and $\pi_1(G)$.

Question: For an affine algebraic group $G$,how could one calculate $\pi_1(G)$? in particular what are $\pi_1(GL_r)$, $SL_r$? and $Sp(r)$?

Thanks in advance!

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    $\begingroup$ $SL_r$ and $Sp_r$ are simply connected. The determinant induces an isomorphism $\pi _1(GL_r)\stackrel{\sim}{\rightarrow }\pi _1(\mathbb{G}_m)=\widehat{\mathbb{Z}}$. $\endgroup$
    – abx
    Commented May 4, 2015 at 20:48

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Over $\mathbb{C}$, the étale fundamental group is the profinite completion of the topological fundamental group. You certainly assume that your affine algebraic group $G$ is connected. The question reduces easily to the case when $G$ is reductive. For $G$ reductive, the topological fundamental group of $G$ is computed in M. Borovoi, Abelian Galois cohomology of reductive groups, Memoirs of the AMS 132 (1998), No. 626. See Section 1, especially Definition 1.3 and Proposition 1.11.

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    $\begingroup$ When I apply the prop 1.11 to the case $G=GL_r$, I got $\pi_1(GL_r)=Hom(\mathbb Z,\mathbb Z)=\mathbb Z$ But As @abx said, $\pi_1(GL_r)=\hat{\mathbb Z}$ ?? what is the problem here? $\endgroup$
    – Z.A.Z.Z
    Commented May 10, 2015 at 10:24
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    $\begingroup$ @AllyMath: Prop. 1.11 gives the topological fundamental group $\mathbb{Z}$, then you should take the profinite completion in order to get the étale fundamental group $\widehat{\mathbb{Z}}$. $\endgroup$
    – Mikhail Borovoi
    Commented May 11, 2015 at 14:04

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