Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n1.Supported by DGICYT:PS90-0226     

Tensor products and homotopies for ω-groupoids and crossed complexes

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves non-abelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ω-groupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.     

G-symmetric spectra,semistability and the multiplicative norm

In this paper we develop the basic homotopy theory of G-symmetric spectra (that is, symmetric spectra with a G-action) for a finite group G, as a model for equivariant stable homotopy with respect to a G-set universe. This model lies in between Mandell's equivariant symmetric spectra and the G-orthogonal spectra of Mandell and May and is Quillen equivalent to the two. We further discuss equivariant semistability, construct model structures on module, algebra and commutative algebra categories and describe the homotopical properties of the multiplicative norm in this context.     

Topology of Hom complexes and test graphs for bounding chromatic number

The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on the chromatic number. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, ⊙) and Hom(⊙, G) complexes as functors from graphs to posets, and introduce a functor (⊙)¹ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of Hom complexes in terms of spaces of equivariant poset maps and Γ-twisted products of spaces. When P:= F(X) is the face poset of a simplicial complex X, this provides a useful way to control the topology of Hom complexes. These constructions generalize those of the second author from [17] as well as the calculation of the homotopy groups of Hom complexes from [8].     

Homotopy coherent category theory

This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.

     

Localization with respect to a class of maps II — Equivariant cellularization and its application

We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.     

On relations between gradient and classical equivariant homotopy groups of spheres

We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.     

Homotopical intersection theory, II: equivariance

This paper is a sequel to (Klein and Williams in Geom Topol 11:939–977, 2007). We develop here an intersection theory for manifolds equipped with an action of a finite group. As in Klein and Williams (2007), our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. We give applications of our theory to embedding problems, equivariant fixed point problems and the study of periodic points of self maps.     

Equivariant eilenberg maclane spaces K (G, μ, 1) for possibly non-connected or empty fixed point sets

Equivariant Eilenberg-MacLane spaces are constructed in [1, p. II.13], [3, p. 277], [8, p. 45], however, only for nonempty connected H-fixed point sets for all HG and in the pointed category. This is a reasonable assumption in equivariant homotopy theory (equivariant Posnikov-systems, homology, obstruction theory) but too restrictive for the study of equivariant manifolds. Therefore we develope a treatment of equivariant Eilenberg-MacLane spaces of type one in full generality. They are used, for example, in equivariant L-theory as reference spaces (see [5]) or in [4].     

Galois Theory and a New Homotopy Double Groupoid of a Map of Spaces

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat¹-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.     

紧黎曼对称空间到Grassmann流形的等变等距极小浸入

在本文中，我们利用李群及其表示理论作为主要工具，　讨论了紧黎曼对称空间到Ｇｒａｓｓｍａｎｎ　流形的等变等距极小浸入问题．     

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.     

G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index are in a one-one correspondence to the -cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.     

A Hopf type classification theorem for isovariant maps from free <Emphasis Type="Italic">G</Emphasis>-manifolds to representation spheres

An isovariant map is an equivariant map preserving the isotropy subgroups. In this paper, we develop an isovariant version of the Hopf classification theorem; namely, an isovariant homotopy classification result of G-isovariant maps from free G-manifolds to representation spheres under a certain dimensional condition, the so-called Borsuk-Ulam inequality. In order to prove it, we use equivariant obstruction theory and the multidegree of an isovariant map.     

Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups. Received: 12 February 2001; in final form: 15 August 2001 / Published online: 28 February 2002     

Localization,completion and detecting equivariant maps on skeletons

In the category of equivariant spaces with base point, we prove the injectivity of the induced map between homotopy sets under some conditions. We study some relations between the localization and the completion. By using these results, we characterize continuous maps which are homotopic on skeletons, and obtain a generalization of the theory of phantom maps.     

Varieties of simplicial groupoids I: Crossed complexes

It is usual to use algebraic models for homotopy types. Simplicial groupoids provide such a model. Other partial models include the crossed complexes of Brown and Higgins. In this paper, the simplicial groupoids that correspond to crossed complexes are shown to form a variety within the category of all simplicial groupoids and the corresponding verbal subgroupoid is identified.     

A projective homotopy theory of crossedG-modules

In this article we extend Hilton's projective homotopy theory of modules (Hilton, 1967) to a homotopy theory of crossed modules, and then reduce some resulting homotopy classification problems to problems in group homology. We also observe that our homotopy theory satisfies the axioms of a Baues fibration category (Baues, 1989).This author would like to thank the University of Cape Town for its hospitality.     

Strong Homotopy Types,Nerves and Collapses

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial G-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces.     

Equivariant acyclic maps

In this paper we apply a recently developed new version of the Bredon-Illman cohomology theory to obtain an equivariant analogue of a result of Kan and Thurston, which implies that a connected CW-complex has the homotopy type of a space obtained by applying the plus construction of Quillen to certain Eilenberg-MacLane spaces.