We show that coefficients of residue formulas for characteristic numbers associated to a smooth toral action on a manifold can be taken in a quotient field This yields canonical identities over the integers and, reducing modulo two, residue formulas for Stiefel Whitney numbers.
Let ξ be an oriented 8-dimensional vector bundle. We prove that the structure group SO(8) of ξ can be reduced to Sp(2) or Sp(2) · Sp(1) if and only if the vector bundle associated to ξ via a certain outer automorphism of the group Spin(8) has 3 linearly independent sections or contains a 3-dimensional subbundle. Necessary and sufficient conditions for the existence of an Sp(2)- structure in ξ over a closed connected spin manifold of dimension 8 are also given in terms of characteristic classes.
In this note we give examples in every dimension of piecewise linearly homeomorphic, closed, connected, smooth -manifolds which admit two smoothness structures with differing spans, stable spans, and immersion co-dimensions. In dimension the examples include the total spaces of certain -sphere bundles over . The construction of such manifolds is based on the topological variance of the second Pontrjagin class: a fact which goes back to Milnor and which was used by Roitberg to give examples...
Let E → W be an oriented vector bundle, and let X(E) denote the Euler number of E. The paper shows how to calculate X(E) in terms of equations which describe E and W.