Orientability of Surfaces and the Fundamental Group [closed]

Question

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given by $i_{\#}([\alpha])=[i(\alpha)]$, where $i$ denotes the inclusion application of $\Sigma$ into $M$. Why $i_{\#}$ not injective implies $TM\Big|_{\Sigma}$ orientable?

Answer 1

Orientability of $TM|_\Sigma$ is equivalent to non-triviality of normal bundle $N\Sigma$. Denote generator of $\pi_1(\Sigma)$ by $\gamma$; if we have $N\Sigma$ is trivial, then $TM|_{i(\gamma)}$ is non-orientable, so $\gamma$ cannot be nullhomotopic.