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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    have added this pointer:

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    and this one:

    • D. M. Segal, On the symplectic cobordism ring, Commentarii Mathematici Helvetici 45, 159–169 (1970) (doi:10.1007/BF02567323)

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    and this couple:

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    I am trying to find out:

    The K3-surface should represent a nontrivial class in Ω 4 Sp\Omega^{Sp}_4, no?

    I am trying to find a reference that would admit this and provide more details.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 25th 2020

    Presumably at some stage we’d split off ’quaternionic cobordism’, e.g., to include the kind of things Laughton speaks about in Chapter 7:

    We begin with some preliminaries on quaternionic cobordism,… (p. 99)

    ’Quaternionic towers’, etc.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 26th 2020

    The K3-surface should represent a nontrivial class in Ω 4 Sp\Omega^{Sp}_4, no?

    At the end of

    • R. E. Stong, Some Remarks on Symplectic Cobordism, (JSTOR)

    it says

    M 4M^4 is a generator of Ω 4 Sp=Z\Omega^{Sp}_{4}= Z,

    where M 4M^{4} is mentioned at the top of p. 432, and M 4rM^{4r} earlier.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2020
    • (edited Nov 26th 2020)

    Thanks for checking.

    But I cannot tell from this if K3 represents a generator, hence if that M 4M^4 is SpSp-bordant to K3K3. I suspect it is, but then why wouldn’t anyone comment on it.

    I know that the bordism class of K3K3 is a non-trivial element in Ω 4 SU\Omega^{SU}_4, e.g. from Novikov, p. 218 (and in Ω 4 Spin\Omega^{Spin}_4, for that matter). I am thinking this ought to imply at least that it’s also non-trivial in Ω 4 Sp\Omega^{Sp}_4, I suppose.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 26th 2020

    When you say K3K 3, you mean what Novikov has as the Kummer surface K 4K^4?

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 26th 2020
    • (edited Nov 27th 2020)

    (Edit this is a little bit confused, take with a grain of salt)

    I think from (3.16) on page 218 one gets that K 4K^4 is a K3. Certainly c 1,c 2c_1,c_2 and χ\chi work out right. The Pontryagin class p 1p_1 is correct, up to sign, and the signature τ\tau is also correct, up to sign.

    So the only issue is the signs on p 1p_1 and τ\tau, and I don’t know if this is just due to normalisation, or if one needs to take the reverse orientation, or what.

    Hmm, no: the Pontryagin classes are independent of the orientation! So it might be something to do with a choice of basis?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 26th 2020

    And ’Kummer surface’ seems to be in some places a synonym for K3, but it’s not quite clear. A smooth quartic in ℂℙ 3\mathbb{CP}^3 is a K3, and Novikov says a “generic” quartic.

    Hmm, but now I read your comment #7 more closely, this was just about Ω 4 SU\Omega_4^{SU}, and extending this to Ω 4 Sp\Omega_4^{S p}. Man, should have slowed down…

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020
    • (edited Nov 27th 2020)

    Thanks for checking. Maybe it can indeed be decided on the characteristic classes.

    More modern (in fact recent) accounts of K3 as representing an element in the SUSU-bordism ring are referenced at MSU, here. (These authors even mention SpSp-bordism, but only to say that it remains mostly unknown.)

    I was vaguely thinking that the identification SU(2)Sp(1)SU(2) \simeq Sp(1) of the K3’s structure group might allow to see how its bordism class behaved as we move from SUSU-bordism to SpSp-bordism. But one needs more info than that.

    (On terminology: It seems that these days “Kummer surface” is mostly used for the singular incarnation of K3 as the orbifold 𝕋 4/ 2\mathbb{T}^4/\mathbb{Z}_2 Usage seems to have been different when Novikov wrote his survey, not sure.)

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 27th 2020

    Words of wisdom, perhaps, from Nige Ray (here):

    Maybe MSp *M S p_{\ast}, is such a problem because SpS p-manifolds admit so few alternative SpS p structures.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2021

    added the remark that

    The canonical topological group-inclusions

    Sp(k)SU(2k)U(2k) Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k)

    (quaternionic unitary group into special unitary group into unitary group) induce ring spectrum-homomorphism of Thom spectra

    MSpMSUMU M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U}

    (from MSp to MSU to MU)

    and hence corresponding multiplicative cohomology theory-homomorphisms of cobordism cohomology theories.

    diff, v11, current

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