Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n-exact functors, which are analogous to Goodwillie’s n-excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt-Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt-Goerss Spectral Sequence on homotopy. We close by giving an identification of the E² term of the GAHSS in the case of 2-exact functors on Moore spaces.     

On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E₁ term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S³.     

THE FUNDAMENTAL GROUP IN FIBRATIONS

A generalized Mayer-Vietoris sequence involving crossed homomorphisms is established and the construction is applied to the homotopy sequence of the CW-pair (X.X¹) to relate the homotopy sequences of (X.X¹) and the fibre bundle F → E → X in low dimensions. If there is a partial cross-section of E → X over X², the classical form, π₁ E ～ π₁ [xtilde] π₁ F as a semidirect product, results. In case there is no extension over X² of any cross-section of the restricted bundle χ:π₂ (x², x¹) → X¹ the corresponding obstruction map X_E:π₂(x²,x¹) → π₁F is non-trivial and in case F → E → X is an SO(n)-bundle (n ≥ 3), χ_E maps into a subgroup of the centre, Z(π₁ F), of order at most 2.     

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.     

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωⁿ⁺¹G→Ωⁿ⁺¹(G/H)→ΩⁿH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H_∗(Ωⁿ(G/H);F₂). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

Homotopy decomposition of a group of symplectomorphisms of S×S

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S²×S² when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

Homology decompositions for p-compact groups

We construct a homotopy theoretic setup for homology decompositions of classifying spaces of p-compact groups. This setup is then used to obtain a subgroup decomposition for p-compact groups which generalizes the subgroup decomposition with respect to p-stubborn subgroups for a compact Lie group constructed by Jackowski, McClure and Oliver.     

Splitting off rational parts in homotopy types

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (see Theorem 21.3 of [L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York-London, 1970]). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say and generalised Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, T-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.     

Shape fibrations and strong shape theory

The notion of shape fibration was introduced by Marde?i? and Rushing. In this paper we use ‘fibrant space’ techniques in strong shape theory to prove that every shape fibration p:E → B of compact metric spaces is contained in a map of fibrant spaces p′:E′→B′ which enjoys a certain lifting property and whose homotopy properties reflect the strong shape properties of the map p. Standard methods for studying Hurewicz fibrations are readily applied to the map p' and in this way we obtain a number of strong shape generalizations of results of Marde?i? and Rushing. We also prove the following theorem which answers a question of Rushing: A shape fibration of compact metric spaces which is a strong shape equivalence is an hereditary shape equivalence. Since the converse was known, this gives a characterization of hereditary shape equivalences.     

Rational homotopy of the (homotopy) fixed point sets of circle actions

In this paper, when G is the circle S¹ and M is a G-space, we study the rational homotopy type of the fixed point set MG, the homotopy fixed point set MhG, and the natural injection MG→MhG.     

Homology of gauge groups associated with special unitary groups

We study the homology of gauge groups associated with principal SU(n) bundles over the four-sphere. After computing the mod p homology of based gauge groups of SU(n) by combined use of the Serre spectral sequence and the Eilenberg-Moore spectral sequence, we compute the modp homology of gauge groups of SU(n) using the Serre spectral sequence for the evaluation fibration.     

Simple homotopy types and finite spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X↘Y of finite spaces induces a simplicial collapse K(X)↘K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K↘L induces a collapse X(K)↘X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.     

A spanning tree cohomology theory for links

In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard–Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the _E3$E_3$-term of their spectral sequence. The main purpose of the present paper is to prove directly that this _E3$E_3$-term is a link invariant. We also give some concrete examples of computation of the invariant.     

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, _A∞$A_{\infty }$ spaces, _E∞$E_{\infty }$ ring spectra, etc.—are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.     

Secondary derived functors and the Adams spectral sequence

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E₃-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E₃-term explicitly by an algorithm.     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.