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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 18th 2011

started projective unitary group with some remarks on its obstruction theory. Also aded a comment with a pointerto this entry to lifting gerbe

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 20th 2011

Just side remark. The contractibility of the unitary group in finite dimensions used is not so trivial isn’t it ? I never knew these things well, but people I think usually invoke this as Kuiper’s theorem. Just for the record, as I am unfortunately out of normal work conditions at the very moment, so I can not delve into resolving this now. Somebody else, or wait for me much later.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 20th 2011

okay, I have added the statement of Kuiper’s theorem explcitly to unitary group

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021
• (edited Sep 2nd 2021)

I have a question on a statement in:

For $G$ a compact Lie group, the authors consider (p. 5, Sec. 1.2) the topological space

$Hom(G, PU(\mathcal{H}))/_{ad} PU(\mathcal{C})$

of continuous group homomorphisms int the projective unitary group on a separable Hilbert space, modulo conjugation – topologized as a quotient space of a subspace of the mapping space with its compact-open topology.

A priori it’s not clear (to me) that or where this space is discrete.

But the authors seem to prove (Prop. 1.6) that the subspace on the “stable” maps (Def. 1.5 on p. 6)

$Hom_{st}(G, PU(\mathcal{H})) \;\subset\; Hom(G, PU(\mathcal{H}))$

yields a discrete quotient

$Hom_{st}(G, PU(\mathcal{H}))/_{ad} PU(\mathcal{C}) \;\; \simeq \;\; Ext(G,S^1) \;\; \in \; Set \,.$

I still need to absorb the argument in more detail, and maybe once i have the following will be clear. But right now I am wondering:

Is this discrete space a disjoint summand of the full space?

If so, is it the maximal discrete disjoint summand?

In other words: Would we discover precisely the need to restrict to stable maps if we were just asking for the discrete component of the original space?