What are examples of nontrivial principal fiber bundles?

Question

I am looking for an example of $G$ principal fiber bundle over a topological space with $G$ a topological group such that is not trivial. In particular, I would be glad to have an example where G is $PSL(n)$, $SL(n)$ $GL(n)$ or another algebraic linear group with the natural zariski topology.

Answer 1

Question: "In particular, I would be glad to have an example where G is PSL(n), SL(n) GL(n) or another algebraic linear group with the natural zariski topology."

Answer: Let $k$ be the field of complex numbers and let $V:=k\{e_0,e_1\}, V^*:=k\{x_0,x_1\}$. The group $G:=SL(V)$ acts on $V$, and if you let $l:=k\{e_0\}$ be the line spanned by $e_0$ you get a closed subgroup $P\subseteq G$ - the "stabilizer subgroup" of $l$. You may construct the quotient $\pi: G \rightarrow G/P$ and there is an isomorphism

$$G/P \cong \mathbb{P}(V^*) \cong \mathbb{P}^1_k.$$

You may cover $\mathbb{P}^1_k$ by the open subschemes $U_i:=D(x_i)$ and you will find that there is an isomorphism

$$\pi^{-1}(U_i) \cong U_i \times_k P.$$

Hence the map $\pi$ is a locally trivial principal fiber bundle in the Zariski topology with fiber $P$. You get a surjection

$$\pi: SL(V) \rightarrow \mathbb{P}(V^*) $$

from an affine scheme $SL(V)$ onto the projective line $\mathbb{P}(V^*)$ with $P$ as "fibers".

Note: The reason this is a "fun and interesting" example is the following: All linebundles $L(d):=\mathcal{O}(d)$ on the projective line $C$, has a canonical $SL(V)$-action. This action induce an action on the global sections $H^0(C,L(d)) \cong Sym^d(V^*)$.

$$\rho:SL(V) \rightarrow GL( Sym^d(V^*))$$

And $H^0(C, L(d))$ is an irreducible $SL(V)$-module. Moreover: All finite dimensional irreducible $SL(V)$-modules may be constructed in this way. This is the famous Borel-Weil-Bott formula.

In general if you let $G:=SL(\mathbb{C}^n)$ and $H \subseteq G$ any closed subgroup, you may construct the quotient

$$\pi: G \rightarrow G/H$$

and $G/H$ is a smooth quasiprojective variety of finite type over $\mathbb{C}$. The map $\pi$ is locally trivial in the etale topology. In fact: This type of example was what led people to introduce the "etale topology". If the group $G$ is semi simple and $H$ is parabolic it follows the quotient map $\pi$ is locally trivial in the Zariski topology.