The cobordism group $N(M^n)$ of codimension-one immersions in the $n$-manifold $M^n$ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group $gr^*N(M^n)$. We introduce some new invariants for immersions enlightening the Atiyah-Hirzebruch spectral sequence associated to $N(M)$, which are of combinatorial-geometric nature. Explicit computations are developed for $n\leq7$, and the group structure is also investigated for orientable 4-manifolds.
Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by na…
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi and A. Ranicki. In the codimension one case, we provide a slightly weaker statement. We also give proofs of rearrangement and cancellation theorems for handles of embedded submanifolds with boundary.
We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k-1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k-1)-space. Moreover, any generic homotopy in (6k-1)-space connecting f to g must have at lea…
For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fol…
Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy group of the infinite dimensional real projective space onto the 2-primary part of the stable homotopy group of spheres). Analogous results - modulo small primes - are obtained for cusp maps and more complicated Morin maps as well.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallel…
The notion of cobordism of singular maps was introduced around 1980 by A. Szűcs and U. Koschorke independently. As an application, Szűcs used it to compute cobordism groups of immersions and embeddings in dimensions where the classical theory did not succeed. His method of studying cobordism groups of singular maps involves the investigation of classifying spaces that were constructed by him, first in a few special cases, then, with the help of the work of Rimányi, in complete generality. Since then quite a few results and (partial) computations were performed in this theory. The present thesi…
Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the relative homotopy principle on the existence level. We also deal with the oriented version and construct a classifying space to which this oriented cobordism group is represented as the set of homotopy classes of P in the codimension n<p and n\geqq p\geqq 2.
Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences that connect the oriented and unoriented (abstract) cobordism groups $Ω_n$ and $\mathfrak{N}_n$. In this paper we present analogous exact sequences connecting the oriented and unoriented cobordism groups of maps with prescribed singularities. This gives positive answer to a fifteen-year-old question posed by Szűcs and has interesting consequences even in the case of cobordisms of immersions.
The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra whose dimensions are equal to those of the target manifolds and which have simplicial structures compatible with (the canonical) simplicial structures of the source and the target manifolds, and in considerable cases they inherit fundamental and important invariants of source manifolds. In fact, Reeb spaces are fundamentall tools in the algebraic and different…
We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands.
We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we a…
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings $M^{2n-1} \subset S^{2n+1}$ under isotopy and cobordism. The second main result (update: which is false) is that the $S$-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi--Levine re…
We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the negative end and such that the positive end is a stabilisation whereas the negative end admits augmentations. These examples emphasise point (i) of the definition of exact Lagrangian cobordisms in [6]. In order to provide such examples we construct Lagrangian immersions with single d…
Recently the author has introduced cobordism-like modules induced from generic maps whose codimensions are negative. They are generalizations of cobordism modules of manifolds. They have been introduced in generalizing the following theorem shown by Hiratuka and Saeki in 2013--14; for a generic map whose codimension is negative including a connected component of an inverse image of a regular value being not null-cobordant and for a space defined as all connected components of inverse images, which is a polyhedron of dimension equal to that of the target space, the top-dimensional homology grou…
The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of ${\bf R}^n$ for l…
The notion of the geometrical $\Z/2 \oplus \Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+ε$-range) admits a geometrical $\Z/2 \oplus \Z/2$--control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e. there exists a mapping $κ_0: M^{\frac{3n+q}{4}} \to \…
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical projection induces the branched covering map. In the case of the 3-sphere, the branch locus is a knotted or linked curve in space, the 2-fold branched cover always embeds, and the 3-fold branch cover might be immersed. In the case of the 4-sphere, the branch locus is a knotted or link…
