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Let $S^1\times S^3$ be parametrised as $\{(\alpha,\beta, \gamma)\in \mathbb{C}^3||\alpha|^2+|\beta|^2=1, |\gamma|=1\}$ and let $S^{1'}=\{e^{i\theta}(1,0,1)|\theta\in [0,2\pi]\}$. I would like to see if the induced homomorphism, $i_*:\pi_1(S^{1'})\to\pi_1(S^1\times S^3)$ is injective.

I have tried to investigate this by defining a homotopy between the loop $\gamma(t)=e^{it}(1,0,1)$ to the basepoint, $(1,0,1)$ but was not able to define such a homotopy.

I would greatly appreciate any advice on whether such a homotopy exists or if there is a more elegant way of approaching this?

Thank you very much

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Recall that $\pi_1(X\times Y)\cong \pi_1(X)\times \pi_1(Y)$ via the projections. Since $S^3$ is simply connected, the projection map $p:S^1\times S^3\to S^1$ induces an isomorphism $p_*:\pi_1(S^1\times S^3)\to \pi_1(S^1)$. But $p\circ i$ is a homeomorphism ${S^1}'\to S^1$, so $i_*p_*$ is an isomorphism, and hence $i_*$ is an isomorphism.

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