On Evaluation Subgroups of Generalized Homotopy Groups

The purpose of this paper is to give a new proof of the Joyal-Tierney theorem (unpublished), which asserts that a morphism f : R → S of commutative rings is an effective descent morphism for modules if and only if f is pure as a morphism of R-modules. Let R be a commutative ring with unit and R−mod the category of R-modules. Since, for any R-module M , the group C(M) = HomAb(M,Q/Z) (where Ab is the category of abelian groups and Q/Z is the rational circle abelian group) becomes an R-module with the action of R on C(M) by (rf)(m) = f(rm) , we can define a functor C : (R−mod)op → R−mod, given by C(M) = HomAb(M,Q/Z). Since the abelian group Q/Z is an injective cogenerator in the category of abelian groups (see, for example, [1]), the functor C is exact and reflects isomorphisms. We say that a morphism f : M → M ′ of R-modules is pure if for any R-module N , 1N ⊗R f : N ⊗R M → N ⊗R M ′ is monic. Let f : M → M ′ be a morphism of R-modules. The next lemma follows from the commutativity of the diagram HomR(C(M), C(M ′)) ≈ HomR(C(M),C(f)) HomR(C(M,C(M))) ≈ C(C(M)⊗R M ′) C(1C(M)⊗Rf) C(C(M)⊗R M), where the vertical morphisms are the canonical isomorphisms. 1. Lemma. Let f : M → M ′ be a morphism of R-modules. The following conditions are equivalent: (a) f is a pure morphism of R-modules. (b) C(f) is a split epimorphism of R-modules. Let f : R → S be a morphism of commutative rings. Recall that a descent datum on an object M ∈ Ob(S−mod) can be described as an S-module morphism θ : M → S⊗RM such that θ makes Received by the editors 1999 August 9 and, in revised form, 2000 February 2. Published on 2000 March 9. 2000 Mathematics Subject Classification: 13C99,18A20,18A30,18A40.

An endomorphism f of an Abelian group A is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f − v idA)(A) is contained in the maximal divisible subgroup of A for some v ∈ Z.

We study the homology of the dual de Rham complex as functors on the category of abelian groups. We give a description of homology of the dual de Rham complex up to degree 7 for free abelian groups and present a corrected version of the proof of Jean’s computations of the zeroth homology group.

Let G-ZFIDCW denote the category of G-spaces having the G-homotopy type of a finitely dominated G-CW complex for a compact Lie group G. Luck [8] has introduced a functor Wa from G-EF^C^ into the category of abelian groups and has realized the equivariant finiteness obstriction as the element w(X) in Wa(X). That is, a finitely dominated G-CW complex X is G-homotopy equivalent to a finite G-CW complex if and only if w(X)=Q. When G is the trivial group, there is an isomorphism from Wa(X) to the reduced projective group KQ(Z[πι(X)]) which sends the element w (X) to the WalΓs finiteness obstruction ([14]). Anderson [1] and Ehrlich [4] have studied a sufficient condition for w(E)= 0 for some fibration E-^B with fiber S. Munkholm, Pedresen [11], Luck [6, 7, 9] and others have studied the transfer map KQ(Z[^(B)])-^K:Q(Z[^(E)]). The purpose of this paper is to get a sufficient condition for w(X)=0 for a S^space X and a finite cyclic group L. We call G-maps /0: YQ-^X and /0: Y4-*X equivalent if there exists a commutative diagram

Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

This chapter is an introduction to module theory in the category \({\mathtt {Sup}}\) of complete lattices and join preserving maps — these are modules on unital quantales. In this context, an interesting class of modules on involutive and unital quantales is induced by irreducible representations of \(C^*\)-algebras. Since \({\mathtt {Sup}}\) is \(*\)-autonomous, there exist constructions in the connection with modules on unital quantales which are not possible in the traditional theory of modules in the category of abelian groups. As an important example, this chapter explains the details of the equivalence between right \({\mathfrak {Q}}\)-modules and join-complete \({\mathfrak {Q}}\)-valued lattices where \({\mathfrak {Q}}\) is a unital quantale. Finally, a sketch of automata in \({\mathtt {Sup}}\) is given showing that every automaton in \({\mathtt {Sup}}\) gives rise to a right module on the free unital quantale generated by the corresponding input alphabet.

Since we have removed 0 + jO from the complex plane, the Nyquist map D × Ω (ℂ \ {0 + j0}, if it exists, could have a nontrivial homotopy class. The set of all possible homotopy classes of all such maps depends essentially on D. This chapter addresses the question as to what homotopy classes can be expected for a given uncertainty space. In case of such uncontractible uncertainty spaces as toris—multichannel phase margin problems—the set of all homotopy classes is quite large. The next question is to determine which of these homotopy classes yield closed-loop stability. Finally, given an uncertainty structure D, it is shown how the loop function L(s) induces a homotopy class of the map.

Matroidal bijections between graphs

In this paper, we propose some new type of weak cyclic multivalued contraction mappings by generalizing the cyclic contraction using the δ-distance function. Several novel fixed point results are deduced for such class of weak cyclic multivalued mappings in the framework of metric spaces. Also, we construct some examples to validate the usability of the results. Various existing results of the literature are generalized.

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups $$\mathbb{Z} \mathord{\left/ {\vphantom {\mathbb{Z} s}} \right. \kern-\nulldelimiterspace} s$$ with s ∈ S, or in the abelian group $$\mathbb{C}{\left[ {S^{{ - 1}} } \right]} = {\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}} \mathord{\left/ {\vphantom {{\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}} \mathbb{Z}}} \right. \kern-\nulldelimiterspace} \mathbb{Z}$$ where $$\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}$$ is the group of fractions of the form $$\frac{z}{s}$$ with s ∈ S. In the first case, for n > 1 the localized category $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category $${\user2{Ho}}{\left( {\mathcal{T}_{{\mathcal{D}_{n} }} S - {\user2{Top}}*} \right)}$$ is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations $$X^{{\mathcal{T}_{n} S}} \to X$$ , $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} \to X$$ obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} \to X$$ is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces $$X^{{\mathcal{T}_{n} S}} $$ , $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} $$ are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S p is generated by a prime p ∈ P one has that for n > 1 the category $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ is equivalent to the product category $$\Pi _{{p \in P}} {\user2{Ho}}{\left( {\mathcal{T}_{n} S^{p} - {\user2{Top}}*} \right)}$$ . If the multiplicative system S is generated by a finite set of primes, then localized category $${\user2{Ho}}{\left( {\mathcal{T}_{{\mathcal{D}_{n} }} S - {\user2{Top}}*} \right)}$$ is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ .

ALGEBRAIC HOMOTOPY THEORY

Pontrjagin classified mappings of a three dimensional sphere into anndimensional complex, where he made use of a new type of product of cocycles. By the aid of the generalized Pontrjagin’s product of cocycles Steenrod enumerated effectively all the homotopy classes of mappings of an (n+1) dimensional complex into annsphere. According to the recent issue of the Mathematical Reviews it is reported that M. M. Postnikov extended Steenrod’s case to the case where an arcwise connected topological space which is aspherical in dimensions less thann, takes place of annsphere. (Postnikov M. M., Classification of continuous mappings of an(n+1)dimensional complex into a connected topological space which is aspherical in dimensions less thann. Doklady Akad. Nauk SSSR (N.S.) 71., 1027-1028, 1950 (Russian. No. proof is given.)) But here in Japan no details are yet to hand. We intend to give a solution to this problem in case wheren>2, and also to give an application concerning the(n+ 3)-extension cocycle.

Degree theory for continuous maps has a long history and has been extensively studied, both from the point of view of analysis and topology. If f ∈ C 0(S n, S n), deg f is a well-defined element of ℤ, which is stable under continuous deformation. Starting in the early 1980s, the need to define a degree for some classes of discontinuous maps emerged from the study of some nonlinear PDEs (related to problems in liquid crystals and superconductors). These examples involved Sobolev maps in the limiting case of the Sobolev embedding; see Sections 2 and 3 below. (Topological questions for Sobolev maps strictly below the limiting exponent have been investigated in [15] and [14].) In these cases, the Sobolev embedding asserts only that such maps belong to the space VMO (see below) and need not be continuous.

Let M n M_n be the connect sum of n n copies of S 2 × S 1 S^2 \times S^1 . A classical theorem of Laudenbach says that the mapping class group Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is an extension of Out ⁡ ( F n ) \operatorname {Out}(F_n) by a group ( Z / 2 ) n (\mathbb {Z}/2)^n generated by sphere twists. We prove that this extension splits, so Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is the semidirect product of Out ⁡ ( F n ) \operatorname {Out}(F_n) by ( Z / 2 ) n (\mathbb {Z}/2)^n , which Out ⁡ ( F n ) \operatorname {Out}(F_n) acts on via the dual of the natural surjection Out ⁡ ( F n ) → G L n ( Z / 2 ) \operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2) . Our splitting takes Out ⁡ ( F n ) \operatorname {Out}(F_n) to the subgroup of Mod ⁡ ( M n ) \operatorname {Mod}(M_n) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M n M_n . Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with ( Z / 2 ) n (\mathbb {Z}/2)^n .