A parametrized version of Gromov's waist of the sphere theorem
A parametrized version of Gromov's waist of the sphere theorem

Benjamin Matschke, IAS
Fine Hall 214
Gromov, Memarian, and KarasevVolovikov proved that any map $f$ from an nsphere to a kmanifold $(n>=k)$ has a preimage $f^{1}(z)$ whose epsilonneighborhoods are at least as large as the epsilonneighborhoods of the equator $S^{nk}$, assuming that the degree of f is even in case $n=k$. We present a parametrized generalization. For the proof we introduce a FadellHusseini type idealvalued index of Gbundles that is quite computable in our situation and we obtain new parametrized BorsukUlam and BourginYangVolovikov type theorems.