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43
Comparison lemmas and applications for diagrams of spaces
 PREPRINT NO. 11, INSTITÜT FUR EXPERIMENTELLE MATHEMATIK
, 1995
"... We provide a "toolkit " of basic lemmas for the comparison of homotopy types of (homotopy) limits of diagrams of spaces over finite partially ordered sets, among them several new ones. In the setting of this paper, we obtain simple inductive proofs that provide explicit homotopy equivalenc ..."
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We provide a "toolkit " of basic lemmas for the comparison of homotopy types of (homotopy) limits of diagrams of spaces over finite partially ordered sets, among them several new ones. In the setting of this paper, we obtain simple inductive proofs that provide explicit homotopy equivalences. (In an appendix we provide the link to the general setting of diagrams of spaces over an arbitrary small category.) We show how this toolkit of old and new diagram lemmas can be used on quite different fields of applications. In this paper we demonstrate this with respect to ffl the "generalized homotopycomplementation formula " by Bjorner [4], ffl the topology of toric varieties (which turn out to be homeomorphic to homotopy limits, and for which the homotopy limit construction provides a suitable spectral sequence), ffl in the study of homotopy types of arrangements of subspaces, where we establish a new, general combinatorial formula for the homotopy types of "Grassmannian " arrangements,
OPERADS, ALGEBRAS, MODULES, AND MOTIVES
"... Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and ..."
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Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, nLie algebras, nbraid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.
The product formula in unitary deformation Ktheory
 KTheory
"... Abstract. For finitely generated groups G and H, we prove that there is a weak equivalence KG ∧ku KH ≃ K(G × H) of kualgebra spectra, where K denotes the “unitary deformation Ktheory ” functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ ∧ku KG in terms ..."
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Abstract. For finitely generated groups G and H, we prove that there is a weak equivalence KG ∧ku KH ≃ K(G × H) of kualgebra spectra, where K denotes the “unitary deformation Ktheory ” functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ ∧ku KG in terms of connective Ktheory and homology of spaces of Grepresentations. 1.
EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES
, 2004
"... Abstract. We construct hyperhomology spectral sequences of Zgraded and RO(G)graded Mackey functors for Ext and Tor over Gequivariant Salgebras (A ∞ ring spectra) for finite groups G. These specialize to universal coefficient and Künneth spectral sequences. ..."
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Abstract. We construct hyperhomology spectral sequences of Zgraded and RO(G)graded Mackey functors for Ext and Tor over Gequivariant Salgebras (A ∞ ring spectra) for finite groups G. These specialize to universal coefficient and Künneth spectral sequences.
COCHAINES QUASICOMMUTATIVES EN TOPOLOGIE ALGEBRIQUE
"... Le but de cet article est de construire sur un ensemble simplicial X des “formes différentielles ” définies sur un anneau commutatif cohérent1 k. Elles permettent de définir une nouvelle structure d’algèbre différentielle graduée2 d*(X) dite quasicommutative, qui est quasiisomorphe à l’algèbre des ..."
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Le but de cet article est de construire sur un ensemble simplicial X des “formes différentielles ” définies sur un anneau commutatif cohérent1 k. Elles permettent de définir une nouvelle structure d’algèbre différentielle graduée2 d*(X) dite quasicommutative, qui est quasiisomorphe à l’algèbre des cochaînes classiques sur X (d’où la terminologie). On montre que cette structure détermine (sous certaines conditions de finitude) le type d’homotopie de X si k = Z. En particulier, les opérations de Steenrod sur la cohomologie de X, ainsi que les groupes d’homotopie de X peuvent s’en déduire par des méthodes standard d’algèbre homologique. Notre travail est donc analogue à celui de D. Quillen [24] et D. Sullivan [28] en homotopie rationnelle, où les algèbres différentielles graduées commutatives jouent un rôle essentiel. Il est aussi intimement lié à celui de M.A. Mandell sur le type d’homotopie à l’aide des E∞algèbres [19], que nous utilisons à la fin de l’article. Cette structure quasicommutative sur l’algèbre d*(X) enrichit considérablement la théorie classique des cochaînes (notée traditionnellement C*(X)), comme nous comptons le montrer de manière sommaire dans cette introduction. Elle consiste à se donner de manière naturelle, pour tout couple d’espaces X et Y, un sous kmodule différentiel gradué d*(X) ⊗ d*(Y) de
Stability phenomena in the topology of moduli spaces. Surveys in Differential Geometry XIV ed
 and ST Yau, International Press (2010
"... The recent proof by Madsen and Weiss of Mumford’s conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spac ..."
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The recent proof by Madsen and Weiss of Mumford’s conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where “stability phenomena ” occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney’s embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer’s stability theorem on the cohomology of moduli space, and the MadsenWeiss theorem, which proves a generalization of Mumford’s conjecture. We also describe Galatius’s recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in
DETECTING KTHEORY BY CYCLIC HOMOLOGY
, 2005
"... Abstract. We discuss which part of the rationalized algebraic Ktheory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. Key words: algebraic Ktheory of group rings, Hochschild homology, cyclic homology, trace maps. ..."
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Abstract. We discuss which part of the rationalized algebraic Ktheory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. Key words: algebraic Ktheory of group rings, Hochschild homology, cyclic homology, trace maps.
DoldKan Type Theorem for ΓGroups
, 1998
"... Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan th ..."
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Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gammagroups. This construction is based on the notion of crosseffects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the nonabelian setup. Finally a DoldKan type theorem for the category of \Gammagroups is proved. In abelian case our theorem claims that the category of abelian \Gammagroups is equivalent to the category of functors Ab\Omega , where\Om
Combinatorics Of Topological Posets: Homotopy Complementation Formulas
, 1998
"... . We show that the well known homotopy complementation formula of Bjorner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian an ..."
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. We show that the well known homotopy complementation formula of Bjorner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, e Gn (R) and exp n (X) which were introduced and studied by V. Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets exp n (S m ) which leads to a negative answer to a question of Vassilev raised at the workshop "Geometric Combinatorics" (MSRI, February 1997). 1. Introduction One of the objectives of this paper is to initiate the study of topological (continuous) posets and their order complexes from the point of view of Geometric Combinatorics. Recall that finite or more generally locally finite partially ordered sets (posets) already occupy one of privileged ...