Rolando Jimenez

A spectral sequence in surgery theory and manifolds with filtrations

In 1978 Cappell and Shaneson pointed out interesting properties of the Browder–Livesay invariants, which are analogous to the differentials of a certain spectral sequence. Such a spectral sequence was constructed by Hambleton and Kharshiladze in 1991. The main step of the construction of the spectral sequence consists in constructing an infinite filtration of spectra, in which, as is well known, only the first two spectra have a clear geometric meaning. In the present paper a geometric interpretation is given to all the spectra of the filtration in the Hambleton–Kharshiladze construction. Surgery obstruction groups for a system of embedded manifolds are introduced, and it is proved that the spectra realizing these groups coincide with the spectra in the Hambleton–Kharshiladze filtration. The algebraic and geometric properties of these groups and their connections with classical surgery theory are described. An isomorphism between these groups and the Browder–Quinn surgery obstruction groups for stratified manifolds is established. The results obtained are applied to the problem of realization of elements of the Wall groups by normal maps of closed manifolds and to the study of the iterated Browder–Livesay invariants.

SPLITTING ALONG A SUBMANIFOLD PAIR

The paper introduces a group LSP of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the LSP- groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery obstruction group of the ambient manifold to the LSP-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some LSP-groups are computed precisely.

An addition theorem for n-fundamental dimension in metric compacta

Abstract We generalize a notion of Freudenthal by proving that each metric compactum X is the inverse limit under an irreducible polyhedral representation of an extendable inverse sequence of compact triangulated polyhedra. The extendability criterion means that whenever X is a closed subspace of a metric compactum Y , then Y is the limit of an inverse sequence of polyhedra where all the bonding maps and triangulations are extensions of the one for X . We apply this to the theory of n -shape by using it to prove an addition theorem for n -fundamental ( n -Fd) dimension. The theorem states that if a metric compactum Z is the union of two closed subspaces X 1 , X 2 with X 0 = X 1 ∩ X 2 and such that dim Z ⩽ n + 1, then n -Fd Z ⩽ max{ n -Fd X 1 , n -Fd X 2 , n -Fd X 0 + 1}.

Free equivariant extensors

Abstract We prove for a finite group G and a compact metric G -space Y that the conditions (1) Y∈LC n−1 ∩C n−1 , and (2) Y∈G - AE(X) , for every normal n -dimensional space X endowed with a free numerable action of the group G , are equivalent. As a corollary we obtain: (A) For the space X endowed with a free action of the finite group G the conditions (1) the space X is normal, dim X⩽n and K(X;G)⩽n+1 ; (2) the space X is normal, dim X⩽n and K(X;G) ; (3) G∗⋯∗G∈G - AE(X) , are equivalent. (B) For a paracompact space X with a free action of the finite group G the inequality K(X;G)⩽ dim X+1 holds.

On linking of cycles in locally connected spaces

Abstract The paper is devoted to studying the linking of cycles with compacta in LC n -spaces and in particular homology Z -sets. The main two consequences of our considerations are the following: (1) It is proved that a k -dimensional polyhedron cannot link a (n−k−1) -dimensional cycle in an n -dimensional Menger manifold. (2) It is proved that a compact set in an ENR is a homology Z -set provided all its points are homology Z -sets.

Fundamental groupoids of digraphs and graphs

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.

On loop extensions satisfying one single identity and cohomology of loops

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities, and in two variables for the case of commutativity. It is considered a large amount of identities. This groups generalize those defined in works of Nishigori [2] and of Jhonson and Leedham-Green [4]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.ABSTRACT In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3] and of Kenneth and Leedham-Green [2]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.