Commuting Homotopy Limits and Smash Products

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.     

Higher topological cyclic homology and the Segal conjecture for tori

We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory.The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology.We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.     

On the homology spectral sequence for topological Hochschild homology

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of -algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower' notation for the Dyer-Lashof algebra.

     

Topological Hochschild Homology

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)HH(R),from the algebraic K-theory of a ring to its Hochschild homology,which can be used to obtain information about K(R) from HH(R).In [1] Bökstedt described a factorization of this tracemap. The intermediate functor THH(HR) is called the topologicalHochschild homology of the Eilenberg–MacLane spectrumHR associated with R, because it is constructed similarly toHochschild homology with the tensor product replaced by thesmash product of spectra.     

Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.     

Autour de la cohomologie de MacLane des corps finis

Résumé Nous décrivons une nouvelle méthode de calcul de la cohomologie de MacLane des corps finis. Cette théorie est intimement reliée aux extensions du groupe additif déjà étudiées par L. Breen et à l'homologie de Hochschild topologique de M. Bökstedt (et donc à la K-théorie stable). Notre approche utilise de manière cruciale l'annulation de la cohomologie de MacLane du corpsF _P , avec pour coefficients l'algèbre symétrique où l'on a inversé le Frobenius. Nous recourons alors à l'analyse des complexes de Koszul et de De Rham en caractéristique non nulle.

---

Summary A new way of computing MacLane cohomology of finite fields is described. Closely related to this theory are L. Breen's extensions du groupe additif and M. Bökstedt's topological Hochschild homology (and so is stable K-theory, hence). Our approach makes essential use of a cancellation result for MacLane cohomology ofF _P with coefficients in the symmetric algebra where the Frobenius has been inverted. We then proceed through an analysis of the Koszul complex and the De Rham complex in non-zero characteristic.

---

---

Oblatum 28-XI-1992 & 3-IX-1993     

Drinfeld modular polynomials in higher rank

We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring Fq[T]. We derive bounds for the coefficients of these polynomials, and compute some explicit examples in the case where q=2, the rank is 3 and the isogenies have degree T.     

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.     

The Albanese map of the Fano surface of a real M-cubic threefold

The restriction to the set of real points of the Albanese map of the Fano surface of a real M-cubic threefold is considered. Some topological properties of this map are proved.     

Symmetric Spectra and Topological Hochschild Homology

A functor is defined which detects stable equivalences of symmetric spectra. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bökstedts original ad hoc definition. In particular, this shows that Bökstedts definition is correct even for nonconnective, nonconvergent symmetric ring spectra.     

Equivariant homotopical homology with coefficients in a Mackey functor

Let M be a Mackey functor for a finite group G. In this paper, generalizing the Dold-Thom construction, we construct an ordinary equivariant homotopical homology theory with coefficients in M, whose values on the category of finite G-sets realize the bifunctor M, both covariantly and contravariantly. Furthermore, we extend the contravariant functor to define a transfer in the theory for G-equivariant covering maps. This transfer is given by a continuous homomorphism between topological abelian groups.We prove a formula for the composite of the transfer and the projection of a G-equivariant covering map and characterize those Mackey functors M for which that formula has an expression analogous to the classical one.     

Orders and actions of branched coverings of hyperbolic links

Let M, M^′ be compact oriented 3-manifolds and L^′ a link in M^′ whose exterior has positive Gromov norm. We prove that the topological types of M and (M^′,L^′) determine the degree of a strongly cyclic covering branched over L^′. Moreover, if M^′ is a homology sphere then these topological types determine also the covering up to conjugacy.     

The bar complex of an E-infinity algebra

The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction.We apply our construction to cochain complexes of topological spaces, which are instances of E-infinity algebras. We prove that the n-th iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the n-fold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X.     

Simplicial ramified covering maps

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simplicial complexes.     

Simploidals sets: Definitions, operations and comparison with simplicial sets

The combinatorial structure of simploidal sets generalizes both simplicial complexes and cubical complexes. More precisely, cells of simploidal sets are cartesian product of simplices. This structure can be useful for geometric modeling (e.g. for handling hybrid meshes) or image analysis (e.g. for computing topological properties of parts of n-dimensional images). In this paper, definitions and basic constructions are detailed. The homology of simploidal sets is defined and it is shown to be equivalent to the classical homology. It is also shown that products of Bézier simplicial patches are well suited for the embedding of simploidal sets.     

Simplicial ramified covering maps

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simplicial complexes.     

k紧优双环网的无限族的构造

双环网是计算机互连网络和通讯系统的一类重要拓扑结构,已广泛应用于计算机互连网络拓扑结构的设计中.利用L形瓦理论,结合中国剩余定理和二次同余方程的性质,给出了不同于参考文献中的任意k紧优双环网的无限族的构造方法,证明了对任意正整数k,若n(t)=3t2 At B,A=1,3,5,对于一定的B＞(k 1)2,均存在正整数t,使得{G(n(t);s(t))}是k紧优双环网的无限族,而且这样的无限族有无穷多类.作为定理的应用,给出了多类新的k紧优双环网的无限族.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.