Alexey Ananyevskiy: The special linear version of the projective bundle theorem

Submission: 2012, May 28

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 h, including Witt groups and MSL[h^{-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.

2010 Mathematics Subject Classification: 14F42, 19G12, 19G99

Keywords and Phrases: special linear orientation, stable Hopf map, Euler class, Witt groups, special linear Grassmannian

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