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 from the unitary Thom spectrum into complex/unitary K-theory, hence a complex orientation of , and the other is 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 and , hence form a commutative square with them. The CF orientation induces an equivalence:Hopkins & Hovey 1995, p. 1-2
The CF orientation does not induce an equivalence:
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 from the symplectic Thom spectrum:
See also
Atiyah–Bott–Shapiro orientations and Ando–Hopkins–Rezk orientation
Literature
References
External links
Conner-Floyd orientation on the nLab