Singular fibers of two-colored maps and cobordism invariants
type: Item
instance of: scholarly article
title: Singular fibers of two-colored maps and cobordism invariants
zbMATH Open document ID: 1154.57028
author: Takahiro Yamamoto
published in: Pacific Journal of Mathematics
publication date: 2008-11-14T00:00:00Z
review text: Let \(f:M\rightarrow N\) be a smooth map between manifolds with \(\dim M+1\geq\dim N.\) The map \(f\) is \textit{two-colorable} if there exists a pair of disjoint non-empty open subsets \(R\) and \(B\) in \(N-f(S(f))\) such that: \(N-f(S(f))=R\cup B\) and \(\overline{R}\cap\overline{B}=\partial R=\partial B=f(S(f)),\) where \(S(f)\) denotes the set of critical points of \(f.\) If \(f\) has no critical values, it is also two-colorable. The pair \((R,B)\) satisfying the condition above is called a \textit{two-color decomposition} or a \textit{coloring} of \(f.\) The pair \((f,\) \((R,B))\) is called a \textit{two-colored} or a \textit{colored map}. In this paper the fibers of two-colored maps are studied. An equivalence relation among the fibers of colored maps and the notion of \textit{two-colored cobordism} between colored maps is introduced. The theory of fibers of colored maps is also developed and the cochain complex of these fibers is contructed. It is proved that cohomology classes of the cochain complex induce two-colored cobordism invariants of colored maps. Finally, colored stable maps of \(n\)-manifolds into 4-manifolds for \(n=3\) and \(n=5,\) are studied. As an application of this theory an earlier theorem of \textit{T. Yamamoto} [J. Math. Soc. Japan 58, No.~3, 721--742 (2006; Zbl 1105.57027)] and a theorem of \textit{A. Szűcs} [Bull. Lond. Math. Soc. 18, 60--66 (1986; Zbl 0563.57015)] are derived.
reviewed by: Charalampos Charitos
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 5366314
zbMATH Keywords: colored map
zbMATH Keywords: Thom map
zbMATH Keywords: stable map
zbMATH Keywords: singular fiber
zbMATH Keywords: bordism
zbMATH Keywords: cobordism
MaRDI profile type: Publication
full work available at URL: pjm.2008.234.379
OpenAlex ID: W2034991177
DOI: 10.2140/PJM.2008.234.379
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description: scientific article; zbMATH DE number 5366314