A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

Abstract

We present a parametrized version of Volovikov's powerful Borsuk–Ulam–Bourgin–Yang type theorem, based on a new Fadell–Husseini type ideal-valued index of G-bundles which makes computations easy. As an application we provide a parametrized version of the following waist of the sphere theorem due to Gromov, Memarian, and Karasev–Volovikov: Any map f from an n-sphere to a k-manifold (n ≥ k) has a preimage f-1(z) whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator Sn-k (if n = k we further need that f has even degree).