Extrapolating an Euler class

Let $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n \leq d\leq 2n-3$. Let $(a_1,...,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,...,a_n)$ has height $n$. Jean Fasel has associated to this row an element $[(J,\omega_J)]$ in the Euler class group $E^n(R)$, with $\omega_J:(R/J)^n\to J/J^2$ given by $(a_1,...,a_{n-1},a_n a_{n+1})$. If $R$ contains an infinite field $F$ then we show that the rule of Fasel defines a homomorphism from $WMS_{n+1}(R)=Um_{n+1}(R)/E_{n+1}(R)$ to $E^n(R)$. The main problem is to get a well defined map on all of $Um_{n+1}(R)$. Similar results have been obtained by Mrinal Kanti Das and MD Ali Zinna, with a different proof. Our proof uses that every Zariski open subset of $SL_{n+1}(F)$ is path connected for walks made up of elementary matrices.Comment: 7 pages, reference update

From Mennicke symbols to Euler class groups

Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class group. We suggest that this is the last map in a longer exact sequence of abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the sequence is an orbit set of unimodular two by n matrices over the ring A. If n is at least four we describe a partially defined operation on two by n matrices. We conjecture that this operation describes a group structure on G if A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.Comment: 11 pages, to appear in the Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry. TIFR, Mumbai, January 4-12, 200

Spherical complexes attached to symplectic lattices

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree g-2 or lower. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.Comment: 18 p; final versio