The special linear version of the projective bundle theorem

Abstract

AbstractA special linear Grassmann variety$\text{SGr}(k,n)$is the complement to the zero section of the determinant of the tautological vector bundle over$\text{Gr}(k,n)$. For an$SL$-oriented representable ring cohomology theory$A^{\ast }(-)$with invertible stable Hopf map${\it\eta}$, including Witt groups and$\text{MSL}_{{\it\eta}}^{\ast ,\ast }$, we have$A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$, and$A^{\ast }(\text{SGr}(k,n))$is a truncated polynomial algebra over$A^{\ast }(pt)$whenever$k(n-k)$is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of$A^{\ast }(\text{BSL}_{n})$in terms of homogeneous power series in certain characteristic classes of tautological bundles.