Singular fibers of two-colored maps and cobordism invariants

Abstract

We introduce two-colored maps of n-manifolds into m-manifolds, develop the theory of their singular fibers, and study these fibers for the case m = 4 and n = 3, 5.

Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

“…The notion of singular fibers of C ∞ maps between manifolds without boundary was first introduced in [7], where classifications of singular fibers of stable maps M → N with (dim M, dim N ) = (2, 1), (3,2) and (4,3) were obtained. Later, singular fibers of stable maps of manifolds without boundary were studied in [7,8,11,12,16,17,18], especially in connection with cobordisms. The first author [7] established the theory of universal complex of singular fibers of C ∞ maps as an analogy of the Vassiliev complex for map germs [6,15].…”

“…This can be used for getting certain cobordism invariants of singular maps. For example, the first author [7] obtained cobordism invariants for stable Morse functions on closed surfaces, and the second author [18] studied the universal complex of singular fibers of two-colored C ∞ maps, computing its cohomology groups. In these theories, for a certain set of singular fibers τ , cohomology classes of the universal complexes of singular fibers of τ -maps provide τ -cobordism invariants for τ -maps.…”