Necessary conditions for finite critical sets. Maps with infinite critical sets (Q519477)

The authors provide necessary conditions on a given map \(f : M^{n+k}\rightarrow N^n\) between two compact manifolds, such that its critical set is finite. The main results are contained in Theorem 2.1, Corollaries 2.2 and 2.3. To illustrate these results, we mention here Theorem 2.1 (b): If \(k=3\), \(M,N\) are orientable, \(N\) is \(3\)-connected, \(\pi_1(M)\) is isomorphic to a direct product \(G\times H\) of two nontrivial groups and \(\pi_2(M)=0\), then \(F=S^1\times \Sigma_g\), where \(\Sigma_g\) is a compact surface of some genus \(g\geq 1\). Here \(F\) is the fiber of the restriction fibration \(M\setminus f^{-1}(B(f))\rightarrow N\setminus B(f)\), and \(B(f)\) is the bifurcation set of the map \(f\). Using the obtained results, the authors present in Section 3 interesting examples of pairs \((M,N)\) of manifolds such that every map \(f : M\rightarrow N\) has infinite critical set. Other papers by the second author directly connected to this topic are [Proc. Am. Math. Soc. 128, No. 11, 3435--3444 (2000; Zbl 0958.57032); Differ. Geom. Appl. 24, No. 6, 579--587 (2006; Zbl 1122.55009)].