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

$M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U}$and hence corresponding multiplicative cohomology theory-homomorphisms of cobordism cohomology theories.

]]>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.

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.)

]]>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…

]]>(**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?

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

]]>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.

]]>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.

]]>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.

]]>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.

]]>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)

and this one:

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

have added this pointer:

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