Low torsion in orientable manifolds? The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is:

Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only in an $m$ manifold which is nonorientable, and hence twisted into itself in some sense.



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*What are some simple proofs of this fact?

*Can an orientable $m$-manifold have $k$ torsion for $k\leq m-2$?


I'm confused. $H_1(\mathbb{R}P^3)=\mathbb{Z}/{2\mathbb{Z}}$, but is orientable (on edit, this answers your question 2).
EDIT: Maybe it refers to the fact that $H^{m-1}(M)$ can only be torsion if the manifold $M$ is non-orientable. It is Corollary 3.28 in Hatcher. 
