### A Decomposability Criterion for Algebraic 2-Bundles on Projective Spaces.

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This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_{2}$–cohomology groups ${H}^{j}\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

We prove a fixed point theorem for Borsuk continuous mappings with spherical values, which extends a previous result. We apply some nonstandard properties of the Stiefel-Whitney classes.

We estimate the characteristic rank of the canonical $k$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,k}$. We then use it to compute uniform upper bounds for the ${\mathbb{Z}}_{2}$–cup-length of ${\tilde{G}}_{n,k}$ for $n$ belonging to certain intervals.

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.