Not available

Não disponívelIn this work a geometric interpretation of the obstructions to the eXtension of functions obtained_ from intersection of functions, is given. Let Mm and Nn be smooth closed manifolds land let V ⊂ M and K ⊂ N be closed submanifolds of same codimension. One of our goals is to give a necessary and sufficient condition for the existence of a smooth map f: M → N, transversal to K, such that V = f-1(K). In Chapter I we obtain conditions for the non existence of the map f. In Chapter III we find some results that guarantee the existence of such a map. For example: if Vm-2 ⊃ Mm is an oriented submanifold homologous to zero in an oriented manifold M, then there exists f: M → Sn such that f Φ sn-2 and Vm-2 = f-1 (Sn-2)

Not available

Não disponívelNot availabl

Not available

Não disponívelIn this work a geometric interpretation of the obstructions to the eXtension of functions obtained_ from intersection of functions, is given. Let Mm and Nn be smooth closed manifolds land let V ⊂ M and K ⊂ N be closed submanifolds of same codimension. One of our goals is to give a necessary and sufficient condition for the existence of a smooth map f: M → N, transversal to K, such that V = f-1(K). In Chapter I we obtain conditions for the non existence of the map f. In Chapter III we find some results that guarantee the existence of such a map. For example: if Vm-2 ⊃ Mm is an oriented submanifold homologous to zero in an oriented manifold M, then there exists f: M → Sn such that f Φ sn-2 and Vm-2 = f-1 (Sn-2)

Remembering Volodymyr Vasilyovich Sharko

We lost Volodymyr Vasilyovich Sharko who passed away unexpectedly at theage of 65 on October 07, 2014 in Kiev. V. Sharko was the Deputy Director onScientic Works at the Institute of Mathematics and a corresponding memberof the National Academy of Sciences of Ukraine

Functions and vector fields on C(CPn)-singular manifolds

Let M2n+1 be a C(CPn) -singular manifold. We study functions and vector fields with isolated singularities on M2n+1. A C(CPn) -singular manifold is obtained from a smooth manifold M2n+1 with boundary in the form of a disjoint union of complex projective spaces CPn boolean OR CPn boolean OR ... boolean OR CPn with subsequent capture of a cone over each component of the boundary. Let M2n+1 be a compact C(CPn) -singular manifold with k singular points. The Euler characteristic of M2n+1 is equal to chi(M2n+1) = k(1 - n)/2. Let M2n+1 be a C(CPn)-singular manifold with singular points m(1), ..., m(k). Suppose that, on M2n+1, there exists an almost smooth vector field V (x) with finite number of zeros m(1), ..., m(k), x(1), ..., x(1). Then chi(M2n+1) = Sigma(l)(i=1) ind(x(i)) + Sigma(k)(i=1) ind(m(i))

On codimensions k immersions of m-manifolds for k=1 and k=m-2

Let us consider M a closed smooth connected m-manifold, N a smooth ( 2m-2)-manifold and f: M -> N a continuous map, with m equivalent to 1( 4). We prove that if f*: H(1)(M; Z(2)) -> H(1)(f(M); Z(2)) is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97-110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m-2, 2m-3 and 2m-4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281-285, 1992)

Cobordism of maps on Z2Z_2-Witt spaces

11 pagesIn this article we study the bordism groups of normally nonsingular maps $f: X \to Y$ defined on pseudomanifolds $X$ and $Y$. To characterize the bordism of such maps, inspired by the formula given by Stong, we give a general definition of Stiefel-Whitney numbers defined on $X$ and $Y$ using the Wu classes defined by Goresky and Pardon and we show that in several cases the cobordism class of a normally nonsingular map $f:X \to Y$ guarantees that these numbers are zero

COINCIDENCES OF FIBREWISE MAPS BETWEEN SPHERE BUNDLES OVER THE CIRCLE

When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as omega-invariants and Nielsen numbers, as well as the related Nielsen theory.In the setting of sphere bundles, a certain degree map deg(B) turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set F of fibrewise homotopy classes of fibrewise maps. We introduce an addition on F, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP