On codimensions \(k\) immersions of \(m\)-manifolds for \(k=1\) and \(k=m-2\)
The authors study the existence of an immersion homotopic to a codimension \(k\) continuous map \(f:M\to N\) of a smooth closed connected \(m\)-manifold \(M\) to a smooth manifold \(N\) (\(k=1\) or \(m-2\)), satisfying the condition that \(f_*:H_{m-k-1}(M;\mathbb Z_2)\to \breve{H}_{m-k-1}(f(M);\mathbb Z_2)\) be injective. In case \(k=m-2\) (Theorem 2.1), \(f\) is homotopic to an immersion if \(m\equiv 1(4)\). In case \(k=1\) (Theorem 2.4), \(f\) is homotopic to an immersion if (1) \(f\) is orientable, (2) \(H^2(M;\mathbb Z)\) is a 2-torsion group, and (3) \(f_1:M\to N\times \mathbb R,\;f_1(x)=(f(x),0),\) is homotopic to an immersion. The last assumption (3) is somewhat strange. The authors seem to be aware of it. In fact, they quote Adachi's sufficient condition for the existence of codimension 2 immersions. The paper complements some results in [\textit{C. Biasi, J. Daccach} and \textit{O. Saeki}, Manuscr. Math. 104, No.~1, 97--110 (2001; Zbl 0993.57011); \textit{U. Koschorke}, Lect. Notes Math. 1350, 188--212 (1988; Zbl 0663.57017); \textit{B. Li} and \textit{G. Li}, Math. Proc. Camb. Philos. Soc. 112, No.~2, 281--285 (1992; Zbl 0772.57034)]. But, there are no examples of the theorems both for \(k=m-2\) and \(k=1\).
scientific article; zbMATH DE number 5346205
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