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In Sullivan's paper Hyperbolic geometry and homeomorphisms (Proc. Georgia Topology Conf., Athens, Ga., 1977) he makes use of a closed hyperbolic almost parallelizable manifold in every dimension. ($M$ is almost parallelizable if $T(M - p)$ is trivial.) This is used to do a hyperbolic version of Kirby's torus trick. The proof is very complicated and proves something a priori much stronger. Sullivan starts with an arbitrary closed hyperbolic manifold $M$ and uses deep work of his with Deligne to show that there is a finite cover of $M$ that is stably (and hence almost) parallelizable.

My question is whether there is a simpler proof of the existence of closed hyperbolic almost parallelizable manifolds, or even a construction thereof. The closest I could find in the literature is a more direct proof by Long and Reid that some finite cover has $w_2 = 0$. (Virtually spinning hyperbolic manifolds. Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 305–313). Note that the question is easy in dimensions $3$ (all orientable $3$-manifolds are parallelizable) and $4$ (all orientable spin $4$-manifolds are almost parallelizable).

Danny Ruberman
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    Sullivan's result was generalized in Okun's thesis, see https://arxiv.org/pdf/math/0112316.pdf. It still uses Deligne-Sullivan but gives many more details. Not sure if a simpler proof exists. My guess is that for most arithmetic hyperbolic manifolds it is unknown if they are stably parallelizable. – Igor Belegradek Nov 30 '21 at 04:05
  • I suspect you know this but for the benefit of others: in https://arxiv.org/abs/1904.12720 the authors construct first examples of orientable non-spin closed hyperbolic manifolds (in each dimension $>3$). I conclude that in most cases it is unknown if an explicit arithmetic hyperbolic manifold is spin. Showing it is stably paralellizable is even harder. – Igor Belegradek Nov 30 '21 at 14:09
  • @IgorBelegradek Thanks for the references. I didn't know about Okun's thesis, but should have remembered Martelli-Riolo-Slavich's result. More details on the argument via Deligne-Sullivan is surely welcome; Sullivan's paper goes through that all very quickly. – Danny Ruberman Nov 30 '21 at 17:07
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    If I may, I suggest you read Okun, not Sullivan. I recall looking at Deligne's paper long time ago and if memory serves, it was an étale cohomology argument, which is beyond my expertise. – Igor Belegradek Nov 30 '21 at 18:45

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