The odd–even invariant of an oriented matroid
This paper introduces the "odd-even invariant" of an oriented matroid. This is a nonnegative integer associated with the oriented matroid which does not change upon reorientation of the oriented matroid. As is shown here, it is also invariant under duality. Additional results are obtained, for uniform oriented matroids.
In the present paper, we study the Sp-module structure of the cokernel of the Johnson homomorphism of the mapping class groups of surfaces. We detect the Sp-irreducible components with highest weight [1^k] (and [k]) in the cokernel. We also show that the multiplicities of them is equal to one.