### A Chain Level Transfer Homomorphism.

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Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection ${\pi}_{E}$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of ${\pi}_{E}$.

We construct a cohomology transfer for n-fold ramified covering maps. Then we define a very general concept of transfer for ramified covering maps and prove a classification theorem for such transfers. This generalizes Roush's classification of transfers for n-fold ordinary covering maps. We characterize those representable cofunctors which admit a family of transfers for ramified covering maps that have two naturality properties, as well as normalization and stability. This is analogous to Roush's...

Sullivan associated a uniquely determined $DGA{|}_{\mathbf{Q}}$ to any simply connected simplicial complex $E$. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space $E$. In case $E$ is the total space of a principal $G$-bundle, ($G$ is a compact connected Lie-group) we associate a $G$-equivariant model ${U}_{G}\left[E\right]$, which is a collection of “$G$-homotopic” $DGA$’s${|}_{\mathbf{R}}$ with $G$-action. ${U}_{G}\left[E\right]$ will, in general, be different from the Sullivan’s minimal model of the space $E$. ${U}_{G}\left[E\right]$ contains the total rational...

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.