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.

  1. What are some simple proofs of this fact?
  2. Can an orientable $m$-manifold have $k$ torsion for $k\leq m-2$?
  • $\begingroup$ If you're interested in manifolds, it might benefit you some to learn a bunch of simple examples of surfaces and 3-manifolds - as you see in the answer below, this falls quickly to a simple example. $\endgroup$ – user98602 May 24 '16 at 14:26

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.

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