Lower Bounds for the Degree of a Branched Covering of a Manifold

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov's problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of $n$-homomorphisms of graded algebras. The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of $n$-sheeted branched coverings of manifolds and, more generally, of Smith–Dold $n$-fold branched coverings. As a corollary, it is shown that the least degree $n$ of a branched covering of the $N$-torus $T^N$ over the product of $k$ $2$-spheres and one $(N-2k)$-sphere for $N\ge 4k+2$ satisfies the inequality $n\ge N-2k$, while the Berstein–Edmonds well-known 1978 estimate gives only $n\ge N/(k+1)$.