Processing math: 100%
Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 nlab noncommutative noncommutative-geometry number-theory object 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2012
    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 10th 2012

    If we have

    πn(O) = Z/2, Z/2, 0, Z, 0, 0, 0, Z for n = 0, …, 7 mod 8,

    and Sp is half a cycle out, don’t we have

    πn(Sp) = 0, 0, 0, Z, Z/2, Z/2, 0, Z for n = 0, …, 7 mod 8?

    So where does the double cover come in? Why isn’t n=3 the first level to co-kill?

    Hmm, but elsewhere I see π1(Sp(2n))=Z, so a double cover is fine. Is it a questioning of stabilizing the Sp(2n)? Is the universal cover of Sp(2n) of any interest?

    (By the way, any reason for using \mathcal{B} on the spin structure page?)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2012

    Sp is half a cycle out,

    Wait, what do you mean by that? Maybe there is some wrong assumption going into this statement. But maybe I am missing something, let me know what you have in mind.

    but elsewhere I see π1(Sp(n))=

    Yes, the maximal compact subgroup of Sp(n) is U(n), and so both have the same homotopy groups.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 10th 2012
    • (edited Jul 10th 2012)

    By half a cycle, I meant this result:

    πk(O)=πk+4(Sp)

    πk(Sp)=πk+4(O),k=0,1,

    What am I getting wrong here? p. 38 of this has π1(Sp(n))=0 in a table called “Homotopy groups of symplectic groups”

    Oh, am I getting confused between Sp(2n,C), Sp(2n,R) and Sp(n)? But according to this Wikipedia page, only Sp(2n,R) has nontrivial fundamental group.

    OK, so you were talking about the latter.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2012

    Right, I should better disentangle my notation. I have been speaking about Sp(2n,) here throughout. It certainly has π1=.

    Let me try to make the notation in the entries more consistent…

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2012

    So I went through a couple of entries and wrote out

      Sp(2n,\mathbb{R})
    

    and

      Mp(2n\mathbb{R})
    

    and

     Ml(n,\mathbb{R})
    

    and so on everywhere. I hope I caught them all, not to leave a mess.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJul 10th 2012

    Urs, I know some references on metaplectic representation, which I would be glad to add to a relevant entry. I am not sure what is your big plan: to consider it under metaplectic group, under metaplectic structure or to create a new entry metaplectic representation. I would be glad to do create it if this fits your picture.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2012

    I’d think that deserves an entry of its own!