Let us mention that in later sections we will define the group B.W I J / for an arbitrary (not necessarily closed) manifold W . At the moment, however, we consider only maps of closed manifolds, just to simplify the exposition.Carefully choosing the set J of singularities, one may derive cobordism groups related to various objects in geometry and topology. Example 1.1 Since a proper submersion is a smooth fiber bundle, the cobordism group of submersions is closely related to diffeomorphism groups of smooth manifolds. It is also known to be related to various infinite loop spaces, moduli spaces of Riemann surfaces, the cobordism category as in Galatius et al [17] A priori J -cobordism groups do not form generalized cohomology theories since, for example, J -cobordism groups are not defined for topological spaces. In the current paper we propose a counterpart of B.W I J / that for a wide range of sets J can be used to compute B.W I J / in the same way as singular cohomology groups H n .W I R/ can be used to compute De Rham cohomology groups H
We classify stably simple reducible curve singularities in complex spaces of any dimension. This extends the same classification of of irreducible curve singularities obtained by V. I. Arnold. The proof is essentially based on the method of complete transversals by J. Bruce et al.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, cat (M #N ) = max{cat M, cat N }. For topological complexity we prove the inequality TC (M #N ) ≥ max{TC M, TC N } for simply connected manifolds.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.