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The special linear version of the projective bundle theorem
Published online by Cambridge University Press: 07 November 2014
Abstract
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A 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.
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