Cubical homotopical algebra and cochain algebras

Basic homotopical algebra is developed in a setting consisting of a cubical monad [G3], i.e. a cylinder endofunctor I, equipped with connections g^–, g⁺: I² I, and— possibly—with symmetries extending the reversion r: II and the interchange s:I²I² of the standard topological case. Our study is mostly concerned with the Puppe sequence of a map f and its comparison with the sequence of iterated homotopy cokernels off. As an application, the homotopy structure of cochain algebras is studied in the present frame, through the cubical co-monad of the path functor P and the left adjoint cylinder functor I.Work partially supported by MURST Research Projects.     

Tate cohomology and periodic localization of polynomial functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S–module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n)_* is independent of choices. Goodwillies general theory says that to any homotopy functor F from S–modules to S–modules, there is an associated tower under F, {P_dF}, such that FP_dF is the universal arrow to a d–excisive functor. Our first main theorem says that P_dFP_d-1F always admits a homotopy section after localization with respect to T(n)_* (and so also after localization with respect to Morava K–theory K(n)_*). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second main theorem which is equivalent to the following: for any finite group G, the Tate spectrum is weakly contractible. This strengthens and extends previous theorems of Greenlees–Sadofsky, Hovey–Sadofsky, and Mahowald–Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. Mathematics Subject Classification (2000) 55P65, 55N22, 55P60, 55P91     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

Equalizers and Flatness Properties of Acts,II

In Comm. Algebra 30 (3) (2002), 1475–1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A_S over a monoid S that are based on the extent to which the functor A_S$\otimes -$ preserves equalizers. In Semigroup Forum 65 (3) (2002), 428–449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act A_S is called weakly equalizer-flat if the functor A_S$\otimes -$ almost preserves equalizers of any two homomorphisms into the left act _SS, and strongly torsion-free if this functor almost preserves equalizers of any two homomorphisms from _SS into the Rees factor act _S(S/Sc), where c is any right cancellable element of S. (The adverb almost signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizer-flat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.     

Reversible linear systems and their versal deformations

Let R be a fixed linear involution (R ²=id) of the spaceR ⁿ . A linear operator M is said to bereversible with respect to R if RM R=M^–1 and infinitesimally reversible with respect to R if M R=–RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R –RV(–t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Koçak.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 33–54, 1991. Original article submitted April 27, 1988.     

Higher ?ech Theory

We introduce a notion of cover of level n for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg–MacLane sheaf, this specializes to computing derived functor cohomology, up to degree n, via simplicial homotopy classes taken along n-covers. Our approach is purely simplicial and combinatorial.     

Projective S-acts and Exact Functors

Let S be a semigroup. In this paper, projective S-acts and exact sequences in S-Act are studied. It is shown that, for a unitary S-act P, the functor Hom(P, –) is exact if and only if P Se for some idempotent e S.The research was supported by the National Natural Science Foundation of China.1991 Mathematics Subject Classification: 20M50     

Tate–Vogel Completions of Half-Exact Functors

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S _n and S ⁿ ), we define F (X)=lim S _n S ⁿ F(X) and F (X)=lim S _n S ⁿ F(X), and call F and F the Tate–Vogel completions of F. We provide several properties of F and F , and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants (F) and (F) of F. If F=Ext _R ⁱ (M, ), then they coincide with Martsinkovsky's -invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.     

On the Singular Homology of One Class of Simply-connected Cell-like Spaces

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called Snake space. In the sequel we introduced the functor SC(−,−) defined on the category of all spaces with base points and continuous mappings. For the circle S ¹, the space SC(S¹, _*){SC(S^1, {_{*}})} is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC(Z, _*){SC(Z, {_{*}})} for any path-connected compact spaces Z.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

Calculus II: Analytic functors

Homotopy functors (for example, from spaces to spaces) are called analytic if, when evaluated on certain n-cubical diagrams, they satisfy certain connectivity estimates. Tools for verifying these estimates include certain generalizations of the triad connectivity theorem. Waldhausen's functor A is analytic. Analyticity has strong consequences, when combined with the concept derivative of a homotopy functor that was introduced in the previous article in this series. In particular, any analytic functor with derivative zero is, in a sense, locally constant.Research partially supported by NSF grant DMS-8806444 and a Sloan Fellowship.     

Groups of classes of pseudohomotopic singular links. I

By a singular link of type (p₁, p₂) in Sⁿ we mean a pair of continuous mappings with disjoint images. In the paper the concept of the pseudohomotopy of singular links is defined, similar to the concept of concordance of classical links, and it is proved that for n>p₂+2 the set of the classes of pseudohomotopic singular links of type (p₁, p₂) in Sⁿ forms an Abelian group with respect to a componentwise connected summation. This group has been obtained in case n2p₂+1–max{n–p₁–2, 0}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 114–124, 1988.     

Spherical multipliers

It is proven in the paper that if functionf(x)L^p(Rⁿ), where 1/p> 1/2 + 1/(n + 1), then the restriction of the Fourier transform f() to the unit sphere S^n–1 lies in L²(S^n–1). As was shown by Fefferman [1], it follows from this that, when > (n –1)/(2(n + 1)), the Riesz-Bochner multiplier acts in LP(Rⁿ) if (n –1–2)/(2n) <1/p < (n + 1 + 2)/(2n).Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 105–112, January, 1978.The author wishes to thank B. S. Mityagin for his attention to this work.     

Mackey and Frobenius Structures on Odd Dimensional Surgery Obstruction Groups

C. T. C. Wall formulated surgery-obstruction groups L _n (Z[G]) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors L _n (Z[–],w|_–) are modules over the Hermitian-representation-ring functor G₁(Z, –) if the orientation homomorphism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups W _n (Z[G], w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor W _n (Z)[–], _–; w|_–) is a w-Mackey functor as well as a module over the Grothendieck–Witt-ring functor GW₀(Z, –), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n.     

Zur Kennzeichnung von Lorentztransformationen in Endlichen Ebenen

Given a commutative field K we define d(A,B):= (a₁–b₁)²–(a₂–b₂)² for A=(a₁,a₂), B=(b₁,b₂) K². Given moreover a fixed k K^O, W. Benz has asked for all mappings : K²K² such that d(A,B)=k implies. This paper gives an answer if K=GF(p), p=5,7,11: must be a bijective collineation in case p = 7,11; there are non-injective mappings in case p=5. To obtain some of these results we have mads use of a computer.

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Prof. R. Artzy zum 70. Geburtstag gewidmet.     

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

Elliptic semiplanes and group divisible designs with orthogonal resolutions

In this paper we prove that there exists an elliptic semiplaneS(v, k, m) withk –m 2 if and only if there exists a group divisible design GDD _k ((k –m)(k – 1);k –m; 0, 1) withm pairwise orthogonal resolutions. As an example of this theorem, we construct an elliptic semiplaneW(45, 7, 3) and show thatW is isomorphic to the elliptic semiplaneS(45, 7, 3) given by R. D. Baker.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

G-symmetric products of the sphereS 2 and other closed surfaces

It is shown that a functor of G-symmetric degree maps closed 2-manifolds into manifolds if and only if it is isomorphic to the Cartesian product of functors of symmetric degree. We study the canonical stratification of the pseudomanifolds , where M is a closed orientable surface. We find the torsion subgroups of certain homology groups H_i(SP _G ⁿ M,).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 9–14.     

Nielsen numbers and fixed points of mappings of bouquets of circles

In this note we construct maps of a bouquet of circles into itself, for which the familiar lower bound for the number of fixed points in terms of the Nielsen number is extremely ineffective. Precisely, we prove the following theorem: if n 1 and if f: S¹vS¹-S¹vS¹ is a map such that the homomorphism of the fundamental group induced by it carries the canonical generators and , respectively, into 1 and (gb^–1gb^–1)^n–1, then the Nielsen number of the map f is equal to 0, and any map homotopic to f has not less than 2n–1 fixed points.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 135–136, 1982.