# Can one squash the space into the plane without squashing?

## Marianna Csörnyei

created by paolini on 26 Apr 2001

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.

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