TITLE: Non-orientable 3-manifold has infinite fundamental group
QUESTION [20 upvotes]: I'm doing past papers for a first course in algebraic topology.
The question is:

Let $M$ be a 3-dimensional, closed, connected, non-orientable manifold. Show that $M$ has infinite fundamental group.

Is there any way of answering this question without simply quoting a classification theorem for 3-manifolds with finite fundamental group?

REPLY [19 votes]: $\def\QQ{\mathbb Q}$If $\pi_1(M)$ is finite, $H_1(M;\QQ)=0$. If $M$ is non-orientable, $H_3(M;\QQ)=0$. So $\chi(M)=h_0(M;\QQ)-h_1(M;\QQ)+h_2(M;\QQ)-h_3(M;\QQ)=1+h_2(M;\QQ)>0$.
But by Poincaré duality, any odd-dimensional manifold has zero Euler characteristic.