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