On codimensions \(k\) immersions of \(m\)-manifolds for \(k=1\) and \(k=m-2\)
type: Item
instance of: scholarly article
title: On codimensions \(k\) immersions of \(m\)-manifolds for \(k=1\) and \(k=m-2\)
zbMATH Open document ID: 1149.57040
author: Carlos Biasi
author: Alice K. M. Libardi
published in: Manuscripta Mathematica
publication date: 2008-09-23T00:00:00Z
review text: 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 \(k=1\).
reviewed by: Tsutomu Yasui
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 55S35
zbMATH DE Number: 5346205
zbMATH Keywords: immersion
zbMATH Keywords: obstruction theory
MaRDI profile type: Publication
full work available at URL: s00229-008-0193-8
OpenAlex ID: W2043559596
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DOI: 10.1007/S00229-008-0193-8
description: scientific article; zbMATH DE number 5346205