10 may 2001

**Abstract.**

Marianna Csörnyei (University College, London) "Can one squash the space into the plane without squashing? (Lipschitz quotient maps between finite dimensional spaces)" Dipartimento di Matematica - Sala dei Seminari

Giovedi' 10 Maggio 2001 - Ore 18.00 Abstract: A map $f:X\to Y$ between metric spaces $X$ and $Y$ is called a Lipschitz quotient, if there are constants $C$, $D$ for which $B(f(x),Dr)\subset f(B(x,r))\subset B(f(x),Cr))$ holds for every $x\in X$ and $r>0$. The question whether a Lipschitz quotient map between finite dimensional Euclidean spaces can increase the co-dimension of a subspace was answered negatively in dimensions at most two. Here as a warmup we show that for a Lipschitz quotient map $f:R^3\to R^2$ the inverse image of a point cannot be a plane. Then we construct a Lipschitz quotient map $f:R^3\to R^2$ for which the inverse image of a point contains a plane.