Carlos Biasi

On the Betti number of the image of a codimension-

Let f M -N be a codimensionk immersion with normal crossings of a closed m-dimensional manifold. We show that f is an embedding if and only if the (m k + 1 )-th Betti numbers of M and f(M) coincide, under a certain condition on the normal bundle of f.

k

Abstract Let f: Mn−1 → Nn be an immersion with normal crossings between closed connected manifolds. The article is concerned with the problem of separation of N by f(M). The main result of this paper is a converse of the Jordan-Brouwer Theorem, under the hypothesis that M is oriented and H 1 (N; Z 2 ) = 0 . More precisely, with the above hypothesis, f is an embedding if and only if N − f(M) has two connected components.

immersion with normal crossings

Let/: M -• N be a continuous map of a smooth closed ra-dimensional manifold into a smooth ^-dimensional manifold with k = n — m>0. In [2] we have defined a primary obstruction Θ1(f)eHm_k(M;Z2) to the existence of a homotopy between /and a topological embedding. This homology class is represented by the closure of the self-intersection set of a generic smooth map (in the sense of Ronga [9]) homotopic to / and it has been shown that it is a homotopy invariant (for a precise definition of θx(f\ see §2). Thus, if / is homotopic to a topological embedding (not necessarily locally flat), then θ^f) necessarily vanishes. (Nevertheless, we warn the reader that the vanishing of this primary obstruction does not necessarily imply the existence of a homotopy between / and a topological embedding.) In this paper, we study the bordism invariance of the primary obstruction 0i(/), which is a homotopy invariant. Here, two continuous maps/and g.M-+Noϊ a closed m-dimensional manifold M into a manifold N are said to be bordant, if there exist a compact (unoriented) (m + l)-dimensional manifold W with dW the disjoint union of two copies Λ/\ and M2 of M and a continuous map F.W-+N (called a bordism between / and g) with F | M 1 = / a n d F\M2—g (see [4] for the terminology). Note that, here, the domains of / and g are the same manifold. Our main result of this paper is the following.

A note on separation properties of codimension-1 immersions with normal crossings

Abstract Let f : M n −1 → N n be an immersion with normal crossings from a compact connected ( n − 1)-manifold M into a connected, open or compact n -manifold N , where M and N can have boundaries. In this paper, we give a necessary and sufficient condition for f to be an embedding using the number of connected components of N − f ( M ). We also obtain an estimate from the above for the number of connected components of N − f ( M ) for f with only double points as its self-intersection points.

On bordism invariance of an obstruction to topological embeddings

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps.

A remark on the separation by immersions in codimension 1

Let f: M -1 N be a cr map between cr manifolds (r > 1) and K a Cr manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of cr maps g: K -1 N which satisfy a certain transversality condition with respect to f in a weak sense. As an application, by considering the case where K is a point, we obtain some new results about the topological properties of f(Rq(f)), where Rq(f) is the set of point;s of M where the rank of the differential of f is less than or equal to q. In particular, we show a result about the topological dimension of f(Rq(f)), which is closely related to a conjecture of Sard concerning the Hausdorff measure of f(Rq(f)).

The implicit function theorem for continuous functions

Let f :M!N and g :K!N be generic differentiable maps of compact manifolds without boundary into a manifold such that their intersection satisfies a certain transversality condition. We show, under a certain cohomological condition, that if the images f.M/andg.K/ intersect, then the .vC 1/th Betti number of their union is strictly greater than the sum of their .vC 1/th Betti numbers, wherevD dimMC dimK dimN. This result is applied to the study of coincidence sets and fixed point sets.

ON TRANSVERSALITY WITH DEFICIENCY AND A CONJECTURE OF SARD

In this work we present a generalization of an exact sequence of normal bordism groups given in a paper by H. A. Salomonsen (Math. Scand. 32 (1973), 87-111). This is applied to prove that if h : M n → X n+k ,5 n< 2k, is a continuous map between two manifolds and g : M n → BO is the classifying map of the stable normal bundle of h such that (h, g)∗ : Hi(M,Z2) → Hi(X × BO,Z2 )i s an isomorphism for i<n − k and an epimorphism for i = n − k, then h bordant to an immersion implies that h is homotopic to an immersion. The second remark complements the result of C. Biasi, D. L. Goncalves and A. K. M. Libardi (Topology Applic. 116 (2001), 293-303) and it concerns conditions for which there exist immersions in the metastable dimension range. Some applications and examples for the main results are also given.