Homomorphisms of homotopy coalgebras

In this paper we consider the theory of higher order homotopy coalgebras as a collection of spaces between co-H-spaces and suspensions, which dualizes Stasheff's theory of Ak-spaces when these are defined through Ak-structures. Moreover we extend two Berstein-Hilton theorems which deal with the primitive homotopy type of a suspension and the class of a suspension map, respectively.     

Primary decomposition of the J-groups of complex projective and lens spaces

We determine the decomposition of J-groups of complex projective and lens spaces as a direct-sum of cyclic groups.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Configuration space integrals and Taylor towers for spaces of knots

We describe Taylor towers for spaces of knots arising from Goodwillie-Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that certain combinations of integrals, indexed by trivalent diagrams, yield cohomology classes of the stages of the tower, just as they do for ordinary knots.     

Stably extendible tangent bundles over lens spaces

The purpose of this paper is to study the stable extendibility of the tangent bundle τn(p) over the (2n+1)-dimensional standard lens space Ln(p) for odd prime p. We investigate for which m the tangent bundle τn(p) is stably extendible to Lm(p) but is not stably extendible to Lm+1(p), where we consider m=∞ if τn(p) is stably extendible to Lk(p) for any k?n, and determine m in the case n?p−3.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

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.     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

Crossed modules as homotopy normal maps

In this note we consider crossed modules of groups (N→G, G→Aut(N)), as a homotopy version of the inclusion N⊂G of a normal subgroup. Our main observation is a characterization of the underlying map N→G of a crossed module in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call “normal maps” between simplicial groups.     

A Borsuk-Ulam type theorem for sphere bundles over stunted projective spaces

A CW complex B is described as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. For n>1, we determine all positive integers m for which the stunted projective space is I-trivial, where F=R,C or H.     

The genus and the category of configuration spaces

In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly p-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii-Schwarz, and the equivariant Lyusternik-Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced.     

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L-S category as its ‘once punctured submanifold’ N\{P},P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.     

Classification of homotopy Wall's manifolds

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a Wall's manifold (defined as a mapping torus of a Dold manifold), introduced by Wall in his 1960 Annals paper on cobordism, have been done by determining: (1) the normal invariants of Wall's manifolds, (2) the surgery obstruction of a normal invariant and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy Wall's manifolds (to be made precise in the body of the paper). Consequently classification results of automorphisms (self homeomorphisms, and self PL-homeomorphisms) of Dold manifolds follow.     

Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {a_n}_n=0^∞ of F. We give asymptotic formulas for {a_n}_n=0^∞ and for F^(k)(z) near the support of μ, hence the precise growth rates on |a_n| and |F^(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

Homology rings of homotopy associative H-spaces

Let X be a homotopy associative mod p H-space for p an odd prime. The homology H_*(X;Fp) is an associative ring, but not necessarily commutative. We study conditions when for elements of H_*(X;Fp). Under certain conditions imply for l=p−2 or p−1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44-50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495-509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E₇, Proc. Amer. Math. Soc. 131 (10) (2003) 3289-3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476-485] to nonfinite mod p homotopy associative H-spaces.     

Tangent circle bundles admit positive open book decompositions along arbitrary links

In the present paper we generalize the divide lying in the unit disk, introduced by A'Campo, to compact, oriented, smooth surfaces, and prove a fibration theorem for generalized divides. As a consequence, we will show that, for any link L in the tangent circle bundle Y to the compact surface, there exists an additional knot K such that the link L∪K is the binding of a “positive” open book decomposition of Y.     

On trivialities of Stiefel-Whitney classes of vector bundles over highly connected complexes

It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel-Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel-Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces.