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added a bit to framed manifold
fixed a silliness at framed manifold, due to me, which somebody kindly pointed out by email (the alleged proof that every orientable 3-manifold admits a framing had really been the trivial argument that every spin 3-manifold admits a framing, instead of the non-tautological argument that $w_2$ also vanishes) fixed now.
after the example of the parallelizable sphereas I added the remark on division algebras and pointers to Adam’s original proof.
added brief remark (here) on relation to intersection pairing and Kervaire invariant.
did a bunch of little edits today on the $n$Lab while the $n$Formum was out of business. Now I don’t have the energy to announce them all. There were a few related to framings.
I added to framed manifold some results about which left invariant framings on compact simple simply-connected Lie groups give nontrivial elements in stable homotopy groups of spheres.
There is a conjecture that for semisimple compact connected Lie groups with rank bigger than some fixed constant, all of the associated stable homotopy elements are zero. The guess for the constant is at or below 10, and the known facts show that for simple such Lie groups, at rank 4 and above this is true.
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