Extrapolating an Euler class

Let R be a noetherian ring of dimension d and let n   be an integer so that n≤d≤2n−3n≤d≤2n−3. Let (a1,…,an+1)(a1,…,an+1) be a unimodular row so that the ideal J=(a1,…,an)J=(a1,…,an) has height n  . Jean Fasel has associated to this row an element [(J,ωJ)][(J,ωJ)] in the Euler class group En(R)En(R), with ωJ:(R/J)n→J/J2ωJ:(R/J)n→J/J2 given by (a¯1,…,a¯n−1,a¯na¯n+1). If R contains an infinite field F   then we show that the rule of Fasel defines a homomorphism from WMSn+1(R)=Umn+1(R)/En+1(R)WMSn+1(R)=Umn+1(R)/En+1(R) to En(R)En(R). The main problem is to get a well defined map on all of Umn+1(R)Umn+1(R). Similar results have been obtained by Das and Zinna [5], with a different proof. Our proof uses that every Zariski open subset of SLn+1(F)SLn+1(F) is path connected for walks made up of elementary matrices.