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     

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.     

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.     

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid arises from the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engström to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that this process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.     

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.     

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.     

The Unstable Equivariant Fixed Point Index and the Equivariant Degree

A correspondence between the equivariant degree introduced byIze, Massabó, and Vignoli and an unstable version ofthe equivariant fixed point index defined by Prieto and Ulrichis shown. With the help of conormal maps and properties of theunstable index, a sum decomposition formula is proved for theindex and consequently also for the degree. As an application,equivariant homotopy groups are decomposed as direct sums ofsmaller groups of fixed orbit types, and a geometric interpretationof each summand is given in terms of conormal maps.     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.     

From graphs to ortholattices and equivariant maps

A functor is constructed from the category of graphs and graph homomorphisms to the category of spaces with involutions and equivariant homotopy classes of maps. This can sometimes be used to prove lower bounds on chromatic numbers, and was inspired by Lovász's proof of Kneser's conjecture. Ortholattices occur as an intermediate step between graphs and spaces, and the correspondence between graphs and ortholattices is analyzed.     

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 the push-forwards for motivic cohomology theories with invertible stable Hopf element

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.     

Gromov convergence of pseudoholomorphic disks

The goal of this paper is to give a self-contained exposition of Gromov compactness for pseudoholomorphic disks in compact symplectic manifolds. The proof leads naturally to the concept of stable maps which was first introduced by M. Kontsevich. Our definition of stable maps for disks is based on the one given by D. McDuff and D. Salamon for spheres. We also generalize the notion of Gromov convergence to the case of disks. We show that the homotopy class is preserved under Gromov convergence. Dedicated to Vladimir Arnold     

Homotopy classification of maps between r-1 connected 2r dimensional manifolds

In this paper,we study the homotopy classification of continuous maps between two r-1 connected 2r dimensional topological manifolds M,N.If we assume some knowledge on the homotopy groups of spheres,then the complete classification can be obtained from the homotopy invariants of M,N.We design an algorithm and compose a program to give explicit computations.     

Monoidal Uniqueness of Stable Homotopy Theory

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, -spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence.     

Long Box Bracket Operations in Homotopy Theory

Secondary homotopy operations called box bracket operations were defined in the homotopy theory of an arbitrary 2-category with zeros by Hardie, Marcum and Oda (Rend Ist Mat Univ Trieste, 33:19–70 2001). For the topological 2-category of based spaces, based maps and based track classes of based homotopies, the classical Toda bracket is a particular example of a box bracket operation and subsequent development of the theory has refined, clarified and placed in this more general context many of the properties of classical Toda brackets. In this paper, and for the topological case only, we use an inductive definition to extend the theory to long box brackets. As is well-known, the necessity to manage higher homotopy coherence is a complicating factor in the consideration of such higher order operations. The key to our construction is the definition of an appropriate triple box bracket operation and consequently we focus primarily on the properties of the triple box bracket. We exhibit and exploit the relationship of the classical quaternary Toda bracket to the triple box bracket. As our main results we establish some computational techniques for triple box brackets that are based on composition methods. Some specific computations from the homotopy groups of spheres are included.     

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.     

Derived Functors of Graded Theories

The homotopy category of one point unions of spheres and the homotopy category of Eilenberg–MacLane spaces are examples of graded theories. Spaces yield models of these theories which can be described in terms of iterated comma categories given by functors. The associated derived functors play an important role in spectral sequences computing homotopy groups. Using cross-effect methods, the derived functors are studied and the existence of a vanishing line is proven.     

Stable splittings of surface mapping spaces

We study the homotopy type of mapping spaces from Riemann surfaces to spheres. Our main result is a stable splitting of these spaces into a bouquet of new finite spectra. From this and classical results, one may deduce splittings of the configuration spaces of surfaces.