Zero divisors of atomic functions (Q1197344)

The authors develop a theory of zero divisors or zero currents for sections of a vector bundle under an orientation condition. An \(R^ n\)- valued function is said to be atomic if the function pulls back the basic forms \(dy^ I/| y|^ p\) on \(R^ n\) to locally Lebesgue integrable forms on the domain manifold in the range \(p=| I| \leq n-1\). Such a function automatically has a zero divisor or zero current. By considering vector bundles the concept of an atomic section is introduced. The main result is that the notion of zero divisor is independent of the choice of local frame. The introduction of the notion of an atomic section and its zero divisor is done by showing that under very mild geometric conditions a smooth \(R^ n\)-valued function is automatically atomic. Section 4 includes results about the exact nature of the divisors of an atomic section. Finally, in section 5 the authors prove that the divisor of an atomic section of a vector bundle \(E\), when considered as a cohomology class in \(H^ n(M,Z)\), is the Euler class of the bundle \(E\).