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Two utterly unrelated questions
Posted:
Jul 18, 1996 11:20 AM


1) One thing I know about triality is that Out(so(8,C)) = S_3. In other words, the group of automorphisms of so(8,C), modulo inner automorphisms, is S_3. But: can this be lifted to a homomorphism from S_3 to Aut(so(8,C))? Also: can this be done in a way that gives automorphisms fixing so(8), the compact real form  so that we get a homomorphism from S_3 to Aut(so(8))?
(Duality for sl(n,C) has these nice properties, so why not triality?)
2) There's often a nice map from H^n(X) to H^{n1}(Loops(X)), where Loops(X) is the space of unbased loops in the topological space X. I think sometimes people call it "transgression". If the lower cohomology groups of X are trivial (zero except for H^0(X) = Z), is this an isomorphism? Okay, I'll admit it: I'm really only interested in the case when X = G is a compact simple Lie group and n = 3, or X = BG and n = 4, or X = Loops(G) and n = 2. But I have the feeling there is some Hurewicz theorem/spectral sequence/etc. argument here I should learn about.
Please Cc: replies to me, since reading the news from where I am now is a pain.



