On two results about fibrations.
On two results of Singhof
For a compact connected semisimple Lie group we shall prove two results (both related with Singhof’s paper [13]) on the Lusternik-Schnirelmann category of the adjoint orbits of , respectively the 1-dimensional relative category of a maximal torus in . The techniques will be classical, but we shall also apply some basic results concerning the so-called -category (cf. [14]).
On uncountable collections of continua and their span
We prove that if the Euclidean plane contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.
On universality of countable and weak products of sigma hereditarily disconnected spaces
Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power of any subspace X ⊂ Y is not universal for the class ₂ of absolute -sets; moreover, if Y is an absolute -set, then contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute -set, then contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power of...
On weak homotopy
On weak -homotopy equivalences of modules
Si definisce il gruppo di —omotopia di un singolo modulo e si introduce la nozione di equivalenza -omotopica debole. Sotto determinate condizioni per l'anello di base oppure per i moduli considerati, le equivalenze -omotopiche deboli coincidono con le equivalenze -omotopiche (forti).
On weak monomorphisms and weak epimorphisms in the category of pro-groups.
In this paper we characterize weak monomorphisms and weak epimorphisms in the category of pro-groups. Also we define the notion of weakly exact sequence and we study this notion in the category of pro-groups.
On Whitehead's inequality, .
On Yetter's invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups.
On Δ-spaces and fundamental dimension in the sense of Borsuk
On -rings and topological realization.
Once More on the Lefschetz Fixed Point Theorem
An abstract version of the Lefschetz fixed point theorem is presented. Then several generalizations of the classical Lefschetz fixed point theorem are obtained.
One codimensional Wecken type theorems.
One-parameter Fixed Point Theory.
Open subgroups of free topological groups
The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.
Open subsets of fibrations
Open/Closed string topology and moduli space actions via open/closed Hochschild actions.
Opérades cellulaires et espaces de lacets itérés
L’espace des configurations de points distincts de admet une filtration naturelle qui est induite par les inclusions des dans . Nous caractérisons le type d’homotopie de cette filtration par les propriétés combinatoires d’une structure cellulaire sous-jacente, étroitement liée à la théorie des -opérades de May. Cela donne une approche unifiée des différents modèles combinatoires d’espaces de lacets itérés et redémontre les théorèmes d’approximation de Milgram, Smith et Kashiwabara.
Operads for -ary algebras – calculations and conjectures
In [8] we studied Koszulity of a family of operads depending on a natural number and on the degree of the generating operation. While we proved that, for , the operad is Koszul if and only if is even, and while it follows from [4] that is Koszul for even and arbitrary , the (non)Koszulity of for odd and remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.
Operads in iterated monoidal categories.