### A non-linear version of Swan's theorem.

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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 ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$ 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 ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$ whose Zariski closure in ${\mathbb{P}}^{n}\left(\u2102\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$.

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.

We present short direct proofs of two known properties of complete flat manifolds. They say that the diffeomorphism classes of m-dimensional complete flat manifolds form a finite set ${S}_{CF}\left(m\right)$ and that each element of ${S}_{CF}\left(m\right)$ is represented by a manifold with finite holonomy group.

This paper contains a description of various geometric constructions associated with fibre bundles, given in terms of important algebraic object, the “twisting cochain". Our examples include the Chern-Weil classes, the holonomy representation and the so-called cyclic Chern character of Bismut and others (see [2, 11, 27]), also called the Bismut’s class. The later example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to the Witten’s index formula....

We exhibit a six dimensional manifold with a line bundle on it which is not the pullback of a bundle on ${S}^{2}$.

We study secondary obstructions to representing a line bundle as the pull-back of a line bundle on ${S}^{2}$ and we interpret them geometrically.