The question of which manifolds admit a free involution seems natural enough, yet I couldn't find much about it online. It's not hard to see that such a manifold must be a boundary, but as pointed out in the answer here that is not enough. I'm wondering whether there are any nice characterizations of these manifolds, hopefully with an algebraic topological flavor (so e.g. is this condition preserved under homotopy equivalence?)
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$\begingroup$ I suggest to look at the book of Conner and Floyd. $\endgroup$ – Thomas Rot Jan 18 '17 at 7:34
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2$\begingroup$ When the manifold has a contractible cover, this implies that the fundamental group has a torsion-free two-fold extension. $\endgroup$ – YCor Jan 18 '17 at 8:06
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$\begingroup$ @YCor -- and this condition is also sufficient for aspherical 3-manifolds, it turns out. $\endgroup$ – HJRW Jan 18 '17 at 9:40
