Let $k$ be an algebraicly closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution. My questions are:

• How could one calculate the fundamental group of $SL_n(k)^\sigma$ ? (the invariant subgroup)

• In particular, what is $\pi_1(SO_n(k))$ and $\pi_1(Sp_{2n}(k))$?

thanks

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution. My questions are:

• How could one calculate the fundamental group of $SL_n(k)^\sigma$ ? (the invariant subgroup)

• In particular, what is $\pi_1(SO_n(k))$ and $\pi_1(Sp_{2n}(k))$?

thanks

Let $k$ be an arbitrary algebraicly closed field. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution. My questions are:

• How could one calculate the fundamental group of $SL_n(k)^\sigma$ ? (the invariant subgroup)

• In particular, what is $\pi_1(SO_n(k))$ and $\pi_1(Sp_{2n}(k))$?

thanks

Let $k$ be an algebraicly closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution. My questions are:

• How could one calculate the fundamental group of $SL_n(k)^\sigma$ ? (the invariant subgroup)

• In particular, what is $\pi_1(SO_n(k))$ and $\pi_1(Sp_{2n}(k))$?

thanks