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

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).