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For which positive integers $n$ can $\mathbb R P^n$ be given the structure of a topological group?

I believe that $\mathbb R P^n$ cannot be given a Lie group structure for even $n$, since then it is not orientable. But, this doesn't necessarily imply it doesn't have a topological group structure (which is not smooth); moreover, it tells us nothing about odd $n$. And ideas?

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    It does for $n=\infty$.2012-06-23

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As Olivier has soon this can be reduced to the question for spheres.

One way to prove this is to note that for a topological group $G$ we have that $G$ is homotopy equivalent to $\Omega BG$, the loopsapce of the classifying space. For example $\Omega BS^1 = \Omega \mathbb{C}P^\infty = \Omega K(\mathbb{Z},2) = K(\mathbb{Z},1) = S^1$. Thus the question is which spheres are loop spaces of classifying spaces. Adams' work showed this is true only for $n=0,1,3$.

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    It should be noted that Adams was concerned with when the spheres are $H$-spaces (amongst a raft of other equivalent conditions). He showed that this is only true for $n=0,1,3,7$. Note a topological group is always a $H$-space, but the converse can fail to be true. The $H$-space structure on $S^7$ does not give a group structure since the octonion's are not associative under multiplication2012-06-22