Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted [[projective unitary group]] with some remarks on its obstruction theory. Also aded a comment with a pointerto this entry to [[lifting gerbe]]

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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeApr 20th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexJust 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 20th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I have added the statement of Kuiper's theorem explcitly to [[unitary group]]

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 2nd 2021
   • (edited Sep 2nd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have a **question** on a statement in: * [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061)) 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](https://arxiv.org/pdf/1202.1880.pdf#page=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? <a href="https://ncatlab.org/nlab/revision/diff/projective+unitary+group/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/projective+unitary+group/6">v6</a>, <a href="https://ncatlab.org/nlab/show/projective+unitary+group">current</a>

   I have a question on a statement in:

   • Noé Bárcenas, Jesús Espinoza, Michael Joachim, Bernardo Uribe, Universal twist in Equivariant K-theory for proper and discrete actions, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) (arXiv:1202.1880, doi:10.1112/plms/pdt061)

   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?

   diff, v6, current