In the theory of spectra and generalized (co)homology theory in mathematics, the Atiyah–Bott–Shapiro orientations (short ABS orientations) are special spectrum morphisms. One is from the spin spectrum into real/orthogonal K-theory and the other is from the spinᶜ spectrum into complex/unitary K-theory. These represent the  genus and the Todd genus. The ABS orientations are named after Michael Atiyah, Raoul Bott und Arnold Shapiro, who introduced it in 1963.
Properties
The ABS orientations are compatible with the canonical spectrum morphisms and , hence form a commutative square with them. The ABS orientation induces an equivalence:Hopkins & Hovey 1995, Theorem 1
The ABS orientation induces an equivalence:
The ABS orientation is surjective on homotopy groups, hence induces surjections .
See also
Conner–Floyd orientations and Ando–Hopkins–Rezk orientation
Literature
References
External links
Atiyah-Bott-Shapiro isomorphism on the nLab