Zbigniew Fiedorowicz

Homology of classical groups over finite fields and their associated infinite loop spaces

Infinite loop spaces associated with ImJ.- Permutative categories of classical groups over finite fields.- K-theory of finite fields and the ImJ spaces.- Calculations at the prime 2.- Calculations at odd primes.- The homology of certain finite groups.- Detection theorems at the prime 2.- Detection theorems at odd primes.- Homology operations associated with the classical groups.

Iterated monoidal categories

Abstract We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Bergers complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgrams small model for Ω 2 Σ 2 X when n=2 and is very closely related to Milgrams model of Ω n Σ n X for n>2.

Classifying Spaces of Topological Monoids and Categories

0. Introduction. In recent years, a number of curious and seemingly paradoxical properties of the bar constuction have come to light. These results have the general form that, in certain situations, the bar construction on a topological group, monoid, or category can be largely independent of the topological structure of the underlying object. Among the most prominent of these results is that of Thurston [28] which states that if G = Homeo(M) is the topological group of self-homeomorphisms of a compact manifold and G& is the same group endowed with the discrete topology, then BG6 -+ BG is a homology equivalence (cf. also MacDuff [16]). In a similar spirit there are two rather amazing results, due to KanThurston [14] and MacDuff [15], that given any connected CW complex X there is a discrete group G and a homology equivalence BG -+ X and a discrete monoid M and a homotopy equivalence BM X. Most recently Friedlander and Milnor have conjectured that for any Lie group G, BGBG is a homology equivalence with finite coefficients. This paper analyzes in detail one particular class of these phenomena, the case when BM6 -+ BM or BC 6 -+ BC is a weak homotopy equivalence for a topological monoid M or a topological category C. The authors interest in this sort of phenomenon arose in trying to understand Waldenhausens work on the algebraic K-theory of spaces (cf. [30]). If X is a connected space, Waldhausen considers the topological category C,,k of G-equivalences of spaces having the homotopy type of G + A V kSn, where G is the Kan loop group of X. He then defines the algebraic K-theory A(X) in various ways, among them (1) A(X) = Z X limnk(BC6 ,k) and (2) A(X) = Z X lim nk(BCn,k)+. Here Cn,k is the discrete category obtained from Cn,k by discarding the topology on the function spaces of Cn,k. Both of these definitions play an important role in Waldhausens theory: (1) is required to compare A(X) with Hatchers higher simple homotopy theory;

On the multiplicative structure of topological Hochschild homology

We show that the topological Hochschild homology THH.R/ of an En ‐ring spectrum R is an En 1 ‐ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the little n‐cubes operad Cn is an EnC1 ‐operad, a result which is of independent interest. 55P43; 18D50

A note on the spectra of algebraic K-theory

Let us denote by the composite map of spectra Here go: X0+X0 is understood to be the identity map. On the n-th space level the map t,, and the structural equivalence of the spectrum X, yield a map A straightforward diagram chase demonstrates the commutativity of the diagrams (up to sign)

Power maps and epicyclic spaces

Abstract In this paper, we present the basic facts of the theory of epicyclic spaces for the first time considered by T. Goodwillie in an unpublished letter to F. Waldhausen. An epicyclic space is a space-valued, contravariant functor on a category ʌ which contains the cyclic category. Our results parallel basic facts of the theory of cyclic spaces established in [1] and [7]. We show that the geometric realization of an epicyclic space has an action of a monoid which is a semidirect product of S 1 and the multiplicative monoid of natural numbers. We also show that the homotopy colimit of an epicyclic space is homotopy equivalent to the bar construction for the monoid action. Finally, we give an explicit description of the homotopy type of the classifying space of the category ʌ.

Volodin K-theory of A∞-ring spaces

IN [IS], Suslin showed that a certain simplicial set Y(G, R), constucted by Volodin, was weakly equivalent to R(BG(R)+), where G(R) = CL(R), E(R) or St(R), and R is a discrete ring with unit. This model for the loop space of K-theory has had a number of important applications, ranging from the stability problem in algebraic K-theory Cl63 to the construction of invariants for stable pseudio-isotopy theory [ZO]. More recently, it has been used in determining the homotopy-type of n-relative K-theory ([6], [9]), and in showing that the Waldhausen K-theory of various diagrams of simplicial rings can be computed degreewise [6]. The existence of a Volodin model for A, rings is presupposed in a crucial reduction step required to prove the equivalence of Waldhausen’s stable K-theory and Biikstedt’s topological Hochschild homology [lo]. In this paper, we construct a Volodin model for the loop space of K(R), where R is an A, ring. It is worth noting that even in the case of simplicial rings, the proof that the Volodin model has the right homotopy type does not follow directly by the arguments of [lS]. and as far as we are aware of there is no rigorous proof of this fact in the existing literature. A key point in Suslin’s argument for discrete rings is that the fundamental group of the Volodin space acts trivially on the higher homotopy groups. This is not true for simplicial rings. For consider the natural map R + q,R for a simplicial ring. The induced map on Volodin spaces V-Y(R) -+ ^YY(noR) (VU( ) is denoted X( ) in [15]) is an isomorphism in homology (both spaces are acyclic) and on fundamental groups (Sr(q,R) in both cases). St(n,R) acts trivially on the higher homotopy groups of ,V9’(noR). If it acts trivially on the higher homotopy groups of YY(R) too, then 3/Y(R) + YY(noR) would be a nilpotent homology equivalence and hence a homotopy equivalence, which is not true in general. In the ratioal case as well as for simplicial rings with torsion-free no, Song [14] proved that the Volodin model has the right homotopy type using methods of homological algebra to reduce the proof to Suslin’s arguments. For general A, rings, the proof is much harder. For this reason, we have devoted this paper to proving the existence and the relevant properties of our Volodin model, deferring to other papers the applications which provided the initial motivation for the construction of such a model for RK(R). After the introduction of A, rings with involution in Section 1 (which we will need for the proof of our main result) we introduce the Volodin construction in Section 2 and prove our main theorem based on three lemmas. In Section 3 we give alternative models for this construction which we will need for the proofs of the three lemmas in Sections 4, 5, and 6. Finaly,

Rectification of weak product algebras over an operad in at and op and applications

7 is devoted to generalizing the group completion theorem of [4] and casting it in homotopy invariant form. In the process we also simplify a number of arguments in that

Associahedra and weak monoidal structures on categories

We develop an alternative to the May-Thomason construction used to compare operad based infinite loop machines to that of Segal, which relies on weak products. Our construction has the advantage that it can be carried out in

A counterexample to a group completion conjecture of J C Moore

Cat