This work is a contribution to study residues of real characteristic classes of vector bundles on which act compact Lie groups. By using the Cech-De Rham complex, the realisation of the usual Thom isomorphism permites us to illustrate localisation techniques of some topological invariants.

We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^n\left(ℝ\right)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n\ge 6$. If the codimension two submanifold is a nonsingular algebraic subset of ${\mathbb{P}}^n\left(ℝ\right)$ whose Zariski closure in ${\mathbb{P}}^n\left(ℂ\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^n\left(ℝ\right)$.

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with the help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy...

In [R] explicit representatives for $S^3$-principal bundles over $S^7$ are constructed, based on these constructions explicit free $S^3$-actions on the total spaces are described, with quotients exotic $7$-spheres. To describe these actions a classification formula for the bundles is used. This formula is not correct. In Theorem 1 below, we correct the classification formula and in Theorem 2 we exhibit the correct indices of the exotic $7$-spheres that occur as quotients of the free $S^3$-actions described above.