nLab
MSp

Contents

Contents

Idea

By MSpMSp 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 Ω Sp\Omega^{Sp}_\bullet.

Properties

Relation to MUMU

The canonical topological group-inclusions

1Sp(k)SU(2k)U(2k) 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

MFrMSpMSUMU 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))

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

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

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

References

Last revised on January 23, 2021 at 13:10:10. See the history of this page for a list of all contributions to it.