Roots of mappings from manifolds (Q2570007)

The author studies Nielsen root theory for maps \(f:X\to Y\) where \(f\) is a proper map from a connected \(n\)-manifold \(X\) into a space \(Y\) which need not be manifold but which admits a universal covering space. If \(f\) is ``transverse to \(y_0\in Y\)'' (i.e., if \(y_0\) has a neighbourhood which is evenly covered by \(f\)) then the author generalizes \textit{H. Hopf}'s notion of absolute degree [Math. Ann. 102, 562--623 (1929; JFM 55.0965.02)] to this case. Assume in addition that \(y_0\) has a neighbourhood homeomorphic to \(\mathbb{R}^n\) then the author proves that any map which is properly homotopic to \(f\) and transverse to \(y_0\) has at least \(\mathcal{A}(f,y_0)\) roots where \(\mathcal{A}\) denotes the absolute degree. If \(n>2\) there is such a map with precisely \(\mathcal{A}(f,y_0)\) roots. In the same situation, each map properly homotopic to \(f\) has at least \(\text{PNR}(f,y_0)\) roots at \(y_0\) where \(\text{PNR}\) denotes the proper Nielsen root number and every Nielsen root class of \(f\) at \(y_0\) with nonzero multiplicity is properly essential. In case \(n>2\) nonzero multiplicity is equivalent to being properly essential. The author shows that if \(Y\) is not a manifold it may happen that a map have a nonzero Nielsen root number which is strictly smaller than the root Reidemeister number. The article explains all the terms used above and provides careful proofs. The author also explains in detail what is known in the manifold case and compares his results to what was known before. The reviewer rates the article as highly enlightening.