Flat vector bundles and open coverings
  We establish a generic counting formula for the Euler number of a flat vector bundle of rank $2n$ over a $2n$ dimensional closed manifold, in terms of vertices of transversal open coverings of the underlying manifold. We use the Mathai-Quillen formalism to prove our result. 
Huitao Feng and Weiping Zhang
math.DG math.AT