The special linear version of the projective bundle theorem

Abstract

A special linear Grassmann variety SGr(k,n) is the complement to the zero section of the determinant of the tautological vector bundle over Gr(k,n). For a representable ring cohomology theory A(-) with a special linear orientation and invertible stable Hopf map η, including Witt groups and MSL[η^{-1}], we have A(SGr(2,2n+1))=A(pt)[e]/(e^{2n}), and A(SGr(2,2n)) is a truncated polynomial algebra in two variables over A(pt). A splitting principle for such theories is established. We use the computations for the special linear Grassmann varieties to calculate A(BSL_n) in terms of the homogeneous power series in certain characteristic classes of the tautological bundle.

Some misprints corrected, slightly revised notation

References in corpus (1)

• On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory

Cited by in corpus (9)

• Motivic Euler characteristics and Witt-valued characteristic classes
• Chow-Witt rings of Grassmannians
• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Stable operations and cooperations in derived Witt theory with rational coefficients
• Quaternionic Projective Bundle Theorem and Gysin Triangle in MW-Motivic Cohomology
• Invariants of upper motives
• Odd rank vector bundles in eta-periodic motivic homotopy theory
• Motivic Pontryagin classes and hyperbolic orientations
• The geometric diagonal of the special linear algebraic cobordism