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Topic: Two utterly unrelated questions
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john baez

Posts: 460
Registered: 12/6/04
Two utterly unrelated questions
Posted: Jul 18, 1996 11:20 AM
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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^{n-1}(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.

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