nLab MSp

Idea

By $MSp$ one denotes the Thom spectrum for stable Sp(n)-structure, representing what is called “symplectic” or “quaternionic” cobordism cohomology theory. The coefficient ring is, accordingly, the “quaternionic”/“symplectic” bordism ring, usually denoted $\Omega^{Sp}_\bullet$.

Properties

Relation to $MU$

The canonical topological group-inclusions

$$1 \;\subset\; Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k)$$

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

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

(from MFr to MSp to MSU to MU)

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

(e.g. Conner-Floyd 66, p. 27 (34 of 120))

Related entries

• quaternionic oriented cohomology theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

• Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)

• Arunas Liulevicius, Notes on Homotopy of Thom Spectra, American Journal of Mathematics Vol. 86, No. 1 (1964), pp. 1-16 (jstor:2373032)

• Pierre Conner, Edwin Floyd, Section 5 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Robert Stong, Some Remarks on Symplectic Cobordism, Annals of Mathematics Second Series, Vol. 86, No. 3 (Nov., 1967), pp. 425-433 (doi:10.2307/1970608)

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

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

• 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)

• Andrew Baker, Some chromatic phenomena in the homotopy of $MSp$, in: N. Ray, G. Walker (eds.), Adams Memorial Symposium on Algebraic Topology, Vol. 2 editors, Cambridge University Press (1992), 263–80 (pdf, pdf)

• Vassily Gorbounov, Nigel Ray, Orientations of $Spin$ Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf, doi: 10.2977/prims/1195168855)

• Vassily Gorbounov, Nigel Ray, Paul Turner, On the Hopf Ring for a Symplectic Oriented Spectrum, American Journal of Mathematics

  Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 (doi:10.2307/2374960)

• Craig Laughton, Quasitoric manifolds and cobordism theory, Manchester 2008 (pdf, pdf)

• Ivan Panin, Charles Walter, On the algebraic cobordism spectra $MSL$ and $MSp$ (arXiv:1011.0651)