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Let $p:S^1\times S^3\rightarrow S^1\times S^3$ be a covering map with $p(z,y)=(z^3,y)$ and $z\in S^1\subset\mathbb{C}$ and $h:\mathbb{R}P^4\rightarrow S^1\times S^3$. Is there a lift $g:\mathbb{R}P^4\rightarrow S^1\times S^3$ with $pg=h$?

I just noticed that I have to show if $g_*(\pi_1(\mathbb{R}P^4))=g_*(\mathbb{Z}/2)\subset p_ * (\mathbb{Z})=p_*(\pi_1(S^1))$ is true or false.

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just notice that the image of $g_*$ is $0$, as $\mathbb{Z}$ is torsion-free – user8268 Nov 12 '11 at 21:14
for any $a\in\pi_1(\mathbb{R}P^4)$ we have $2a=0$; on the other hand, if $b\in\mathbb{Z}$ s.t. $2b=0$ then $b=0$ (now take $b=g_*(a)$) – user8268 Nov 12 '11 at 21:28
well, the image of $g_*$ is $0$, so... – user8268 Nov 12 '11 at 22:32

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