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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added this pointer: * [[Nigel Ray]], _The symplectic bordism ring_, Volume 71, Issue 2 March 1972, pp. 271-282 ([doi:10.1017/S0305004100050519](https://doi.org/10.1017/S0305004100050519)) <a href="https://ncatlab.org/nlab/revision/diff/MSp/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/MSp/8">v8</a>, <a href="https://ncatlab.org/nlab/show/MSp">current</a>

   have added this pointer:

   • Nigel Ray, The symplectic bordism ring, Volume 71, Issue 2 March 1972, pp. 271-282 (doi:10.1017/S0305004100050519)

   diff, v8, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexand this one: * D. M. Segal, _On the symplectic cobordism ring_, Commentarii Mathematici Helvetici 45, 159–169 (1970) ([doi:10.1007/BF02567323](https://doi.org/10.1007/BF02567323)) <a href="https://ncatlab.org/nlab/revision/diff/MSp/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/MSp/8">v8</a>, <a href="https://ncatlab.org/nlab/show/MSp">current</a>

   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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexand this couple: * [[Stanley Kochman]], _The symplectic cobordism ring I_, Memoirs of the American Mathematical Society 1980, Volume 24, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271)) * [[Stanley Kochman]], _The symplectic cobordism ring II_, Memoirs of the American Mathematical Society 1982 Volume 40, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271)) <a href="https://ncatlab.org/nlab/revision/diff/MSp/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/MSp/8">v8</a>, <a href="https://ncatlab.org/nlab/show/MSp">current</a>

   and this couple:

   • Stanley Kochman, The symplectic cobordism ring I, Memoirs of the American Mathematical Society 1980, Volume 24, Number 271 (doi:10.1090/memo/0271)

   • Stanley Kochman, The symplectic cobordism ring II, Memoirs of the American Mathematical Society 1982 Volume 40, Number 271 (doi:10.1090/memo/0271)

   diff, v8, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am trying to find out: The K3-surface should represent a nontrivial class in $\Omega^{Sp}_4$, no? I am trying to find a reference that would admit this and provide more details.

   I am trying to find out:

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

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

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPresumably 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 26th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>The K3-surface should represent a nontrivial class in $\Omega^{Sp}_4$, no? At the end of * R. E. Stong, _Some Remarks on Symplectic Cobordism_, ([JSTOR](https://www.jstor.org/stable/1970608)) it says >$M^4$ is a generator of $\Omega^{Sp}_{4}= Z$, where $M^{4}$ is mentioned at the top of p. 432, and $M^{4r}$ earlier.

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

   At the end of

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

   it says

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

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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2020
   • (edited Nov 26th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for checking. But I cannot tell from this if K3 represents a generator, hence if that $M^4$ is $Sp$-bordant to $K3$. I suspect it is, but then why wouldn't anyone comment on it. I know that the bordism class of $K3$ is a non-trivial element in $\Omega^{SU}_4$, e.g. from [Novikov, p. 218](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf#page=218) (and in $\Omega^{Spin}_4$, for that matter). I am thinking this ought to imply at least that it's also non-trivial in $\Omega^{Sp}_4$, I suppose.

   Thanks for checking.

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

   I know that the bordism class of $K3$ is a non-trivial element in $\Omega^{SU}_4$, e.g. from Novikov, p. 218 (and in $\Omega^{Spin}_4$, for that matter). I am thinking this ought to imply at least that it’s also non-trivial in $\Omega^{Sp}_4$, I suppose.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 26th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhen you say $K 3$, you mean what Novikov has as the Kummer surface $K^4$?

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

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 27th 2020
   • (edited Nov 28th 2020)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex(**Edit** this is a little bit confused, take with a grain of salt) I think from (3.16) on page 218 one gets that $K^4$ is a K3. Certainly $c_1,c_2$ and $\chi$ [work out right](https://en.wikipedia.org/wiki/K3_surface#Calculation_of_the_Betti_numbers). The Pontryagin class $p_1$ is correct, [up to sign](https://en.wikipedia.org/wiki/Pontryagin_class#Pontryagin_classes_on_a_Quartic_K3_Surface), and the signature $\tau$ is also correct, [up to sign](https://en.wikipedia.org/wiki/K3_surface#Properties). So the only issue is the signs on $p_1$ and $\tau$, 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?

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

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

   So the only issue is the signs on $p_1$ and $\tau$, 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?

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 27th 2020
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexAnd 'Kummer surface' seems to be in some places a synonym for K3, but it's not quite clear. A smooth quartic in $\mathbb{CP}^3$ is a K3, and Novikov says a "generic" quartic. Hmm, but now I read your comment #7 more closely, this was just about $\Omega_4^{SU}$, and extending this to $\Omega_4^{S p}$. Man, should have slowed down...

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

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

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020
    • (edited Nov 27th 2020)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks 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 $SU$-bordism ring are referenced at _[[MSU]]_, [here](https://ncatlab.org/nlab/show/MSU#RelationToCalabiYauManifolds). (These authors even mention $Sp$-bordism, but only to say that it remains mostly unknown.) I was vaguely thinking that the identification $SU(2) \simeq Sp(1)$ of the K3's structure group might allow to see how its bordism class behaved as we move from $SU$-bordism to $Sp$-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 $\mathbb{T}^4/\mathbb{Z}_2$ Usage seems to have been different when Novikov wrote his survey, not sure.)

    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 $SU$-bordism ring are referenced at MSU, here. (These authors even mention $Sp$-bordism, but only to say that it remains mostly unknown.)

    I was vaguely thinking that the identification $SU(2) \simeq Sp(1)$ of the K3’s structure group might allow to see how its bordism class behaved as we move from $SU$-bordism to $Sp$-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 $\mathbb{T}^4/\mathbb{Z}_2$ Usage seems to have been different when Novikov wrote his survey, not sure.)

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 27th 2020
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexWords of wisdom, perhaps, from Nige Ray ([here](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256051231)): > Maybe $M S p_{\ast}$, is such a problem because $S p$-manifolds admit so few alternative $S p$ structures.

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

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

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded the remark that The canonical [[topological group]]-[[subgroup|inclusions]] $$ Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k) $$ ([[quaternionic unitary group]] into [[special unitary group]] into [[unitary group]]) induce [[ring spectrum]]-[[homomorphism]] of [[Thom spectra]] $$ M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U} $$ (from [[MSp]] to [[MSU]] to [[MU]]) and hence corresponding [[multiplicative cohomology theory]]-[[homomorphisms]] of [[cobordism cohomology theories]]. <a href="https://ncatlab.org/nlab/revision/diff/MSp/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/MSp/11">v11</a>, <a href="https://ncatlab.org/nlab/show/MSp">current</a>

    added the remark that

    The canonical topological group-inclusions

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

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

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

    (from MSp to MSU to MU)

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

    diff, v11, current