Lawrence-Sullivan models for the interval

Two constructions of a Lie model of the interval were performed by R. Lawrence and D. Sullivan. The first model uses an inductive process and the second one comes directly from solving a differential equation. They conjectured that these two models are the same. We prove this conjecture here.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε_?(X) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε_?(X) where X is the rationalization of a 1-connected CW-complex of finite type.     

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     

Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory

     

Many homotopy categories are homotopy categories

We show that any category that is enriched, tensored, and cotensored over the category of compactly generated weak Hausdorff spaces, and that satisfies an additional hypothesis concerning the behavior of colimits of sequences of cofibrations, admits a Quillen closed model structure in which the weak equivalences are the homotopy equivalences. The fibrations are the Hurewicz fibrations and the cofibrations are a subclass of the Hurewicz cofibrations. This result applies to various categories of spaces, unbased or based, categories of prespectra and spectra in the sense of Lewis and May, the categories of L-spectra and S-modules of Elmendorf, Kriz, Mandell and May, and the equivariant analogues of all the afore-mentioned categories.     

A counter-example to a conjecture of Félix

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L_X. The radical of L_X, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite.Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dimL_X<∞, i.e., if X is elliptic, then L_X is its own radical, and therefore the total dimension of the radical of L_X in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.     

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     

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.     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.     

Rational homotopy groups of generalised symmetric spaces

Abstract. We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In particular, this gives explicit formulas for the rational homotopy groups of all classical compact symmetric spaces. Received: 18 March 2002 / Published online: 14 February 2003 The author is supported by the {\it DFG Graduiertenkolleg “Mathematik im Bereich ihrer Wechselwirkung mit der Physik”} and is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004     

Rational homotopy of Leibniz algebras

We construct a non-commutative rational homotopy theory by replacing the pair (Lie algebras, commutative algebras) by the pair (Leibniz algebras, Leibniz-dual algebras). Both pairs are Koszul dual in the sense of operads (Ginzburg–Kapranov). We prove the existence of minimal models and the Hurewicz theorem in this framework. We define Leibniz spheres and prove that their homotopy is periodic. Received: 19 September 1997 / Revised version: 23 February 1998     

A note on the homotopy theory of pro-G-spectra

This paper represents a step toward a model structure on pro-spectra in which the weak equivalences are the maps inducing pro-isomorphisms of all pro-homotopy groups. We construct a category in which these weak equivalences are inverted and show that we have not inverted “too much,” in the sense that isomorphic objects still give pro-isomorphic cohomology groups.     

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.     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and L_x its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E² term is the Lie algebra Ext_ULx(Q, L_x), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of L_x and to the Gottlieb group of X.     

Batalin-Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space H_∗ΩnX is an algebra over the homology of the framed n-disks operad H_∗fDn [E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (2) (1994) 265-285; P. Salvatore, N. Wahl, Framed discs operads and Batalin-Vilkovisky algebras, Q. J. Math. 54 (2) (2003) 213-231]. We explicitly determine this H_∗fDn-algebra structure on H_∗(ΩnX;Q). We show that the action of H_∗(SO(n)) on the iterated loop space H_∗ΩnX is related to the J-homomorphism and that the BV-operator on H_∗(Ω²X) vanishes on spherical classes only in characteristic other than 2.