Journal of Homotopy and Related Structures最新文献

The isomorphism of Karoubi–Villamayor K-groups with smooth K-groups for monoid algebras over quasi-stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Fr(acute{mathrm{e}})chet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Fr(acute{mathrm{e}})chet algebras having a properly uniformly bounded approximate unit is established.

We estimate the number of homotopy types of orbit spaces for all free and properly discontinuous cellular actions of groups (Grtimes {mathbb {Z}}^m) and (G_1*_{G_0}G_2). In particular, homotopy types of orbits of ((2n-1))-spheres (Sigma (2n-1)) for such actions are analysed, provided the groups (G_0, G_1, G_2) and G are finite and periodic. This family of groups (Grtimes {mathbb {Z}}^m) and (G_1*_{G_0}G_2) contains properly the family of virtually cyclic groups. The possible actions of those groups on the top cohomology of the homotopy sphere are determined as well.

Porter’s approach is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie. Computations concerning the higher order Whitehead product for spheres and projective spaces are presented as well.

Let (mathcal {E}(X)) be the group of homotopy classes of self homotopy equivalences for a connected CW complex X. We consider two classes of maps, (mathcal {E})-maps and co-(mathcal {E})-maps. They are defined as the maps (Xrightarrow Y) that induce homomorphisms (mathcal {E}(X)rightarrow mathcal {E}( Y)) and (mathcal {E}(Y)rightarrow mathcal {E}(X)), respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an (mathcal {E})-equivalence relation between rationalized spaces (X_{{mathbb Q}}) and (Y_{{mathbb Q}}) as a geometric realization of an isomorphism (mathcal {E}(X_{{mathbb Q}})cong mathcal {E}(Y_{{mathbb Q}})).

We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the crossed modules of groups. For a general setting we give sufficient conditions for the existence of a categorical equivalence between internal groupoids and Whitehead sequences.

We show that if G is a compact Lie group and (mathfrak {g}) is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra (U_q(mathfrak {g})) to the twisted cyclic cohomology of quantum group algebra ({mathcal O}(G_q)). We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podle? sphere ({mathcal O}(S^2_q)) is in the image of this map.

For any pair of categories ((mathsf {C,K})) enriched over the category (mathsf {Gpd}) of groupoids, it is possible to define a strong shape category (SSh(mathsf {C,K})) in such a way that, for (mathsf {C}) the category of topological spaces and (mathsf {K}) its full subcategory of spaces having the homotopy type of absolute neighborhoods retracts for metric spaces, one obtains the strong shape category (SSh(mathsf {Top})), as defined by Marde?i?. We also introduce a new category (SS_{tiny mathsf K}) with the same objects as (mathsf {C}) and morphisms given by suitable pseudo-natural transformations into the category of groupoids. The main result is then that such a category (SS_{tiny tiny mathsf K}) is isomorphic to the strong shape category (SSh(mathsf {C,K})), when (mathsf {C}) is also a proper model category.

There exists a unique natural extension of higher Koszul brackets to every unitary associative algebra in a way that every square zero operator of degree 1 gives a curved (L_{infty }) structure.

Let (mathcal {E}) and (mathcal {S}) be toposes. A geometric morphism ({p:mathcal {E}rightarrow mathcal {S}}) is called pre-cohesive if it is local, essential, hyperconnected and the leftmost adjoint preserves finite products. More explicitly, it is a string of adjoints ({p_! dashv p^* dashv p_* dashv p^!}) such that ({p^*:mathcal {S}rightarrow mathcal {E}}) is fully faithful, its image is closed under subobjects, and ({p_!:mathcal {E}rightarrow mathcal {S}}) preserves finite products. We may also say that (mathcal {E}) is pre-cohesive (over (mathcal {S})). For example, the canonical geometric morphism ({widehat{Delta } rightarrow mathbf {Set}}) from the topos of simplicial sets is pre-cohesive. In general, a pre-cohesive geometric morphism ({p:mathcal {E}rightarrow mathcal {S}}) allows us to effectively use the intuition that the objects of (mathcal {E}) are ‘spaces’ and those of (mathcal {S}) are ‘sets’, that ({p^* A}) is the discrete space with A as underlying set of points and that ({p_! X}) is the set of pieces of the space X. For instance, such a p determines an associated (mathcal {S})-enriched ‘homotopy’ category ({mathbf {H}mathcal {E}}) whose objects are those of (mathcal {E}) and, for each X, Y in (mathbf {H}mathcal {E}), ({(mathbf {H}mathcal {E})(X, Y) = p_!(Y^X)}). In other words, every pre-cohesive topos has an associated ‘homotopy theory’. The purpose of the present paper is to study certain aspects of this homotopy theory. We introduce weakly Kan objects in a pre-cohesive topos. Also, given a geometric morphism ({g:mathcal {F}rightarrow mathcal {E}}) between pre-cohesive toposes (mathcal {F}) and (mathcal {E}) (over the same base), we define what it means for g to preserve pieces. We prove that if g preserves pieces then it induces an adjunction between the homotopy categories determined by (mathcal {F}) and (mathcal {E}), and that the direct image ({g_*:mathcal {F}rightarrow mathcal {E}}) preserves weakly Kan objects. These and other results support the intuition that the inverse image of g is ‘geometric realization’. Also, the result relating g and weakly Kan objects is analogous to the fact that the singular complex of a space is a Kan complex.