I'm trying to justify the following claim I've encountered in a paper but am having no luck. Here's the setup.
Let $F$ be a connected, essential surface (possibly with boundary) in a compact, connected, orientable, irreducible 3-manifold $M$ (possibly with boundary), and suppose that there is a component $C\subset M-F$ such that $\pi_1(C)\to \pi_1(M)$ is an isomorphism.
Claim: $F$ separates $M$. In particular, if $F$ is non-separating, then $\pi_1(M-F) \to \pi_1(M)$ is not surjective. In fact, even $H_1(M-F) \to H_1(M)$ is not surjective.
Any insight would be greatly appreciated!
Note: By essential surface, I mean a bicollared surface $F\subset M$ whose components $F_i$ all satisfy:
- $F_i \not\cong S^2$
- $F_i$ is not boundary-parallel
- $\pi_1(F_i) \to \pi_1(M)$ is injective.