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In Ralph Cohen's notes on the topology of fiber bundles pp.62 he states that, since the space of connections $\mathcal{A}(P)$ (where $P$ is a principal $G$-bundle is affine) it is contractible.

I found in Stack Exchange there is a related old question but the context is rather different. Moreover I wish to see a proof that does not require Zariski topology as I am not asking for a proof of the general statement that any affine space is contractible (by the way is this statement true?).

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    The meaning of "affine space" here is fairly different from its meaning in algebraic geometry. Here it just means it's acted on freely and transitively by a vector space. In particular it has the same homotopy type as a vector space, and vector spaces can be contracted by linear homotopies. – Qiaochu Yuan May 23 '16 at 18:14
  • I see. I think I am confused by these two notions. Thanks a lot for pointing this out Qiaochu! – PhysicsMath May 23 '16 at 19:56

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Let $A$ be an affine space over the topological vector space $V$. I assume $A$ is topologized so that the action of $V$ on $A$ (given by $v \mapsto a_0+v$, $a_0$ some chosen point in $A$) is continuous. Then a contruction on $A$, given some choice of element $a_0 \in A$, is defined by $f_t(a) = a_0 + t(a-a_0)$.

When one says '$A$ is affine over $V$' then the above topological assumptions are usually implicitly made. This is in particular true of the space of connections and $\Omega^1(\mathfrak g_P)$.

  • Thanks a lot for your answer Mike! I think I got confused by the notion of "affine space" in the context of algebraic geometry. – PhysicsMath May 23 '16 at 19:58