The special linear version of the projective bundle theorem

Abstract

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 an SL-oriented representable ring cohomology theory A * (−) with invertible stable Hopf map η, including Witt groups and M SL * , * η , we have A * (SGr(2, 2n + 1)) ∼ = A * (pt)[e]/ e 2n , and A * (SGr(k, n)) is a truncated polynomial algebra over A * (pt) whenever at least one of the integers 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 * (BSL n ) in terms of homogeneous power series in certain characteristic classes of tautological bundles.

Chow–Witt rings of classifying spaces for symplectic and special linear groups

“…Integral Stiefel-Whitney classes have been defined by Fasel in [22], who computed CH • (B GL 1 ) and H • Nis (B GL 1 , I • ). Pontryagin classes appear in the η-local computations of Ananyevskiy [1], but the more complicated Bockstein classes have not been considered before. As pointed out by a referee, the hyperbolic morphism…”

“…Now for X and Y of respective bidegrees (2i, i) and (2j, j), Sq 1 (X) · Sq 1 (Y ) has bidegree (2i + 2j + 2, i + j) and hence vanishes by [27,Theorem 19.3]. This implies that τ Sq 1 (X) · Sq 1 (Y ) = 0, proving (1). Alternatively, this can be derived from [14,Theorem 9.2].…”

“…Proof. We first deal with the relations of type (1). Recall that the map Φ n is by definition W (F )-linear, in particular it will send I(F ) into I(F ).…”

Oriented Schubert calculus in Chow–Witt rings of Grassmannians

“…by means of which the cokernel of β can be identified with the cohomology ring H • (Gr(k, n), W) of the Grassmannians with coefficients in the sheaf of Witt groups W. It has a description completely analogous to the integral cohomology mod torsion of the real Grassmannians. Some of these formulas already appeared in the computations of SL-oriented theories (in particular Witt groups) of Grassmannians in [Ana15]. The exact relation with the computations of twisted Witt groups of Grassmannians in [BC12] will be discussed in Section 8.3.…”

SL-oriented cohomology theories

“…We also improve some results on characteristic classes for SL-oriented cohomology theories obtained in [An15] where it was mostly assumed that the Hopf element η is inverted in the coefficients. Extending the proof of [Lev17, Proposition 7.2] to the general setting of SL-oriented cohomology theories we obtain the following theorem.…”