Singular fibers of two-colored maps and cobordism invariants
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.
scientific article; zbMATH DE number 5366314