The ws-singular cohomology of orbifolds

We define a cohomology with integral coefficients of an orbifold M, which we call the ws-singular cohomology ws-Hq(M) of M.     

The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

The duality theorem between the t-singular homology and the ws-singular cohomology of orbifolds

In the present paper, we prove that for an n-dimensional compact orbifold with an s-homological orientation, the duality of the ws-singular cohomology group and the t-singular homology group holds. The key tools are “the t-modification of the cap product” for giving the duality homomorphism and “the Convex Suborbifold Theorem” for extending the local duality isomorphism to the global one. The duality theorem proved in the present paper is a naturally required consequence of the preceding works of the authors.     

Bordism groups and shape theory

The THOM isomorphism Theorem (1) allows an immediate extension to the strong shape category of compacta: The shape homology $\text{TBG}{}_{\mathrm{n}}\text{(X)}$ with coefficient in a THOM spectrum turns out to be isomorphic to the bordism group $\text{Ω}{}^{\mathrm{G}}{}_{\mathrm{n}}\text{(X)}$ which is defined like but with strong shape morphisms replacing continuous mappings (Theorem 1.2).     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

Examples of cohomology manifolds which are not homologically locally connected

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n?3.     

Group-like structures and the whitehead product in pro-homotopy and shape

The notion of ‘H-space’ is of considerable importance in the homotopy theory of CW-complexes. This paper studies a similar notion in the framework of pro-homotopy and shape theories. This is achieved by following the general plan set forth by Eckmann and Hilton. Examples of shape H-space are also given; it is observed that every compact connected topological monoid is a shape H-space. The Whitehead product is defined and studied in the pro-homotopy and shape categories; and, it is shown that this Whitehead product vanishes on an H-object in pro-homotopy. These results are the natural extension of some well-known classical results in the homotopy theory of CW-complexes.     

An approximation theorem in shape theory

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.     

Onf-vectors and Betti numbers of multicomplexes

A multicomplexM is a collection of monomials closed under divisibility. For suchM we construct a cell complex M whosei-dimensional cells are in bijection with thef _i monomials ofM of degreei+1. The bijection is such that the inclusion relation of cells corresponds to divisibility of monomials. We then study relations between the numbersf _i and the Betti numbers of M. For squarefree monomials the construction specializes to the standard geometric realization of a simplicial complex.This work was supported by the Mittag-Leffler Institute during the Combinatorial Year program 1991–92. The second author also acknowledges support from the Serbian Science Foundation, Grant No. 0401D.     

Intrinsic characterization of Alexander-Spanier cohomology groups of compactifications

In this paper we construct a uniform Alexander-Spanier cohomology functor from the category of pairs of uniform spaces to the category of abelian groups. We show that this functor satisfies all Eilenberg-Steenrod axioms on the category of pairs of precompact uniform spaces, is precompact uniform shape invariant and intrinsically, in terms of uniform structures, describes the Alexander-Spanier cohomology groups of compactifications of completely regular spaces.     

Fundamental groups and finite sheeted coverings

It is well known that for a connected locally path-connected semi-locally 1-connected space X, there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ_d of degree d.The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of ‘fundamental group’ have appeared in the mathematical literature, for instance, the Brown-Grossman-Quigley fundamental group, the ?ech-Borsuk fundamental group, the Steenrod-Quigley fundamental group, the fundamental profinite group or the fundamental localic group.The main result of this paper determines different ‘fundamental groups’ that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X.     

A measure of the deviation from there being fibrations between a pair of compact manifolds

In this paper we first study the topology of certain manifold complements. The obtained results are then used to show that the φ-category (cf. Lusternik-Schnirelmann category) of some pairs of manifolds is infinite. Finally, in the last section the equivariant case is considered, examples of equivariant mappings with infinitely many critical orbits being provided.     

Isomorphic homotopy groups of certain regular sets and their images

In this paper we study the relation between the topology of the set R(f) of regular points and the topology of its image f(R(f)), for some special maps acting between two manifolds M and N. The results are oriented towards negative examples for the inverse problem of deciding whether a given closed subset of the source manifold is a critical set.     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.     

HOMOTOPY CLASSIFICATION OF MAPS BETWEEN PSEUDO-PROJECTIVE PLANES

Abstract

The pseudo-projective plane M(q) is obtained by attaching a two cell to a circle by a map of degree q. We here determine the homotopy classes (M(q), M(w)) and some of their properties in the unbased, based and cellular-preserving cases. In the cellular-preserving case the sets have a near-ring structure for q = w but the addition is lost on passing to the based case. The results, known for q = w = 2, are compared to the known result for ?(M(q)) the group of based self-homotopy equivalence classes.     

Self-maps of the product of two spheres fixing the diagonal

We compute the monoid of essential self-maps of Sn×Sn fixing the diagonal. More generally, we consider products S×S, where S is a suspension. Essential self-maps of S×S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.     

PHANTOM MAPS,COGROUPS AND THE SUSPENSION MAP

ABSTRACT

The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases.

In addition, a study of the map g*: Ph(X,Y₁) → Ph(X,Y₂) induced by a map g: Y₁ → Y₂ of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ? Ph(σX, σY) with Y a grouplike space.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

Self homotopy groups with large nilpotency classes

We demonstrate that for any n>0 there exists a compact connected Lie group G such that the self homotopy group [G,G] has the nilpotency class greater than n, where [G,G] is a nilpotent group for a compact connected Lie group G.