nLab MSpinᶜ

Idea

The cobordism theory for manifolds equipped with (stable) spinᶜ structure.

Properties

Atiyah-Bott-Shapiro orientation

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see at Atiyah-Bott-Shapiro orientation

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Relation to complex K-theory

$M Spin^c$ is related to KU in a variant of the Conner-Floyd isomorphism, via the Atiyah-Bott-Shapiro orientation (Hopkins-Hovey 92, Thm. 1)

Related concepts

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

• Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift volume 210, pages 181–196 (1992) (doi:10.1007/BF02571790, pdf)