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Here $S^n$ is the $n$-sphere. I am trying to show it must not be surjective, but I couldn't arrive at a contradiction. A similar question is to show any $S^2\times S^4\to \mathbb{C}P^3$ also has degree 0.

Any hints towards these sort of quetions would be appreciated.

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Let $f:S^3\rightarrow S^2\times S^1$ be a continuous map. $H^3(S^2\times S^1)$ is generated by $[u].[v]$ where $u$ is a generator of $H^2(S^2)$ and $v$ a generator of $H^1(S^1)$, $f^*([u])=0$ and $f^*([v])=0$, this implies that $f^*([u].[v])=0$.

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