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Assume that $X$ and $Y$ are locally compact and locally doubling metricspaces, which are also generalized $n$-manifolds, that $X$ is locally linearlylocally $n$-connected, and that $Y$ has bounded turning. In this paper,addressing Heinonen's ICM 02 talk, we study the geometry of the branch set$\mathcal{B}_f$ of a quasiregular mapping between metric $n$-manifolds. Inparticular, we show that $\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}$ iscountably porous, as is its image $f\big(\mathcal{B}_f\cap \{x\inX:H_f(x)<\infty\}\big)$. As a corollary, $\mathcal{B}_f\cap \{x\inX:H_f(x)<\infty\}$ and its image are null sets with respect to any locallydoubling measures on $X$ and Y, respectively. Moreover, if either $H_f(x)\leqH$ or $H_f^*(x)\leq H^*$ for all $x\in X$, then both $\mathcal{B}_f$ and$f\big(\mathcal{B}_f\big)$ are countably $\delta$-porous, quantitatively, witha computable porosity constant.  When further metric and analytic assumptions are placed on $X$, $Y$, and $f$,our theorems generalize the well-known Bonk--Heinonen theorem and Sarvas'theorem to a large class of metric spaces. Moreover, our results are optimal interms of the underlying geometric structures. As a direct application, weobtain the important V\"ais\"al\"a's inequality in greatest generality.Applying our main results to special cases, we solve an open problem ofHeinonen--Rickman and an open question of Heinonen--Semmes.

Porosity of the branch set of discrete open mappings with controlled linear dilatation