Invariants for Riemann surfaces covered by the disc and hyperbolic manifolds in general involving minimizing measure of image over homotopy homology classes closed curves maps k-sphere into manifold are investigated. The invariants monotonic under holomorphic mappings strictly certain circumstances. Applications to annular regions ℂ tubular neighborhoods compact totally real submanifolds n , n≥...

In this paper, we study the approximate biprojectivity and biflatness of a Banach algebra A find some relations between theses concepts with ?-amenability ? -contractibility, where is character on A. Among other things, show that ?-Lau product L1(G) ?? A(G) approximately biprojective if only G finite, are group Fourier locally compact G, respectively. We also characterize biflat semigroup algeb...