Conner–Floyd orientation

In the theory of spectra and generalized (co)homology theory in mathematics, the Conner–Floyd orientations (short CF orientations) are special spectrum morphisms. One is $\mathrm{MU}\to \mathrm{KU}$ from the unitary Thom spectrum into complex/unitary K-theory, hence a complex orientation of $\mathrm{KU}$, and the other is $\mathrm{MSU}\to \mathrm{KO}$ from the special unitary Thom spectrum into real/orthogonal K-theory. The CF orientations are named after Pierre Conner and Edwin Floyd, who described it in 1966. It laid the basics for the later generalization by the Landweber exact functor theorem.

Properties

• The CF orientations are compatible with the canonical spectrum morphisms $\mathrm{MSU}\to \mathrm{MU}$ and $\mathrm{KO}\to \mathrm{KU}$, hence form a commutative square with them.
• The CF orientation $\mathrm{MU}\to \mathrm{KU}$ induces an equivalence:

  ${\mathrm{MU}}_{*}(-){\otimes }_{{\mathrm{MU}}_{*}}{\mathrm{KU}}_{*}\simeq {\mathrm{KU}}_{*}(-).$

• The CF orientation $\mathrm{MSU}\to \mathrm{KO}$ does not induce an equivalence:

  ${\mathrm{MSU}}_{*}(-){\otimes }_{{\mathrm{MSU}}_{*}}{\mathrm{KO}}_{*}\simeq {\mathrm{KO}}_{*}(-).$

In 1987, Serge Ochanine described this failure and that it can be saved with various modifications. One possibility is instead using a special spectrum morphism $\mathrm{MSp}\to \mathrm{KO}$ from the symplectic Thom spectrum:

${\mathrm{MSp}}_{*}(-){\otimes }_{{\mathrm{MSp}}_{*}}{\mathrm{KO}}_{*}\simeq {\mathrm{KO}}_{*}(-).$

See also

• Atiyah–Bott–Shapiro orientations $\mathrm{MSpin}\to \mathrm{KO}$ and ${\mathrm{MSpin}}^{\mathrm{c}}\to \mathrm{KU}$
• Ando–Hopkins–Rezk orientation $\mathrm{MString}\to \mathrm{tmf}$

Literature

• Pierre Conner, Pierre; Floyd, Edwin Floyd (1966). "Chapter 5" (in en). The Relation of Cobordism to K-Theories. Lecture Notes in Mathematics. 28. doi:10.1007/BFb0071091. ISBN 978-3-540-03610-4. 
• Hopkins, Michael; Hovey, Mark Hovey (1995). "Spin Cobordism Determines Real K-Theory" (in en). https://ncatlab.org/nlab/files/HopkinsHoveyCobordismK.pdf. 

External links

• Conner-Floyd orientation on the nLab