The special linear version of the projective bundle theorem

TL;DR

This paper extends the projective bundle theorem to special linear Grassmann varieties within certain cohomology theories, providing explicit calculations and a splitting principle.

Contribution

It introduces a special linear version of the projective bundle theorem and computes cohomology of special linear Grassmann varieties and classifying spaces.

Findings

A(SGr(2,2n+1))=A(pt)[e]/(e^{2n})

A(SGr(2,2n)) is a truncated polynomial algebra

Established a splitting principle for these theories

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 \eta, including Witt groups and MSL[\eta^{-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.