Involution action on $H^1(S^1times S^2)$
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I am studying about an action $I^*$ on a de Rham cohomology group $H^1(S^1times S^2)$ induced from an action $Icdot (z,x)=(overline{z},-x) $ where $S^1times S^2subset mathbb{C}times mathbb{R}^3$ . Note that, by Kunneth formula, $$H^1(S^1times S^2)=mathbb{R}.$$ Thus, I want to find a nonzero element in $H^1(S^1times S^2)$ and want to see how $I^*$ acts to the element. And my teacher taught me as below. Let $dtheta in Omega^1(S^1)$ be a generator of $H^1(S^1)=mathbb{R}.$ And let $pi:S^1times S^2rightarrow S^1$ and let $omega=pi^*(dtheta)$ . Then clearly, $domega=0$ so $[omega]$ is nonzero element in $H^1(S^1times S^2)$ . If $iota : S^1times {text{north pole}}hookrightarrow S^1times S^2$ is an embedding, observe that $$picirciota =Id implies iota^*pi^*=Id implies iota^*(omega)=dtheta