Some results on Euler class groups

TL;DR

This paper investigates conditions under which stably free modules over regular domains have unimodular elements, linking this to Euler class groups, and introduces a Whitney class homomorphism in algebraic K-theory.

Contribution

It establishes a criterion for the existence of unimodular elements in stably free modules and defines a new Whitney class homomorphism between Euler class groups.

Findings

P has a unimodular element iff its Euler class is zero in E^n(A).

Defined Whitney class homomorphism w(P): E^s(A) → E^{n+s}(A).

Results hold for regular domains containing an infinite field with 2n ≥ d+3.

Abstract

Let A be a regular domain of dimension d containing an infinite field and let n be an integer with 2n\geq d+3. For a stably free A-module P of rank n, we prove that (i) P has a unimodular element if and only if the euler class of P is zero in E^n(A) and (ii) we define Whitney class homomorphism w(P):E^s(A)\ra E^{n+s}(A), where E^s(A) denotes the sth Euler class group of A for s\geq 1.