On a Conjecture of Yamaguchi and Yokura

Let F⟶E⟶pB be a fibration of simply connected elliptic spaces. Our paper investigates the conjecture proposed by T. Yamaguchi and S. Yokura, states that dim Ker π∗pℚ≤ dim Ker H∗p;ℚ+1. Our goal is to prove this conjecture when F and B satisfy the condition πevenFℚ=πevenBℚ=0. We go also on to establish a well-known conjecture of Hilali for a class of spaces which puts it into the context of fibration.

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝn → X.