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.

Abstract. Ananyevsky has recently computed the stable operations and cooperations of rational Witt theory [An15]. These computations enable us to show a motivic analog of Serre's finiteness result:As an application we define a category of Witt motives and show that rationally this category is equivalent to the minus part of SH(k) Q .

The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V2] is proved in this paper for framed motives stating that a natural map of framed S 1 -spectrais a schemewise stable equivalence, where M f r (X)(n) is the nth twisted framed motive of X. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions of suspension P 1 -spectra Σ ∞ P 1 X + with X a smooth algebraic variety. The Cancellation Theorem for framed motives is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groupswhere ZF(X,Y ) is the group of stable linear framed correspondences in the sense of [GP1]. M Gf r (X ) = (M f r (X ), M f r (X )(1), M f r (X )(2), . . .

We argue that the very effective cover of hermitian K-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological K-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.

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