Connecting rational homotopy type singularities

Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W ^1,p (R ^p+1, N) be the Sobolev space of measurable maps from R ^p+1 into N whose gradients are in L ^p . The restriction of u to almost every p-dimensional sphere S in R ^p+1 is in W ^1,p (S, N) and defines an homotopy class in π _p (N) (White 1988). Evaluating a fixed element z of Hom(π _p (N), R) on this homotopy class thus gives a real number Φ _z,u (S). The main result of the paper is that any W ^1,p -weakly convergent limit u of a sequence of smooth maps in C ^∞(R ^p+1, N), Φ _z,u has a rectifiable Poincaré dual . Here Γ is a a countable union of C ¹ curves in R ^p+1 with Hausdorff -measurable orientation and density function θ: Γ→R. The intersection number between and S evaluates Φ _z,u (S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n _z , depending only on homotopy operation z, such that even though the mass may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n _z ) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R ^m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.     

The homotopy type of rational functions

During the preparation of this work each of the authors was supported by an NSF grant, the second author by an NSF-PYI award, and the first and fourth authors by the S.F.B. 170 in G?ttingen     

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 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     

Cyclic maps in rational homotopy theory

The notion of a cyclic map g:AX is a natural generalization of a Gottlieb element in _n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups.Mathematics Subject Classification (2000): 55P62, 55Q05     

Complete intersections in rational homotopy theory

We investigate various homotopy invariant formulations of commutative algebra in the context of rational homotopy theory. The main subject is the complete intersection condition, where we show that a growth condition implies a structure theorem and that modules have multiply periodic resolutions.     

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.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

Some algebraic constructions in rational homotopy theory

In this note we describe constructions in the category of differential graded commutative algebras over the rational numbers Q which are analogs of the space F(X, Y) of continuous maps of X to Y, the component F(X, Y,ƒ) containing ƒ ε F(X, Y), fibrations, induced fibrations, the space Γ(π) of sections of a fibration π: E → X, and the component Γ(π,σ) containing σ ε Γ (π). As a focus, we address the problem of expressing π^*(F(X, Y, ƒ)) = Hom(π_*(F(X,Y, ƒ)),Q) in terms of differential graded algebra models for X and Y.     

The Samelson product and rational homotopy for gauge groups

This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration ev_p0 :C(P, K) ^K →K, whereC(P, K) ^K is the gauge group of a continuous principalK-bundle. We show that in the case of a bundle over a sphere or a orientable surface the connecting homomorphism is given in terms of the Samelson product. As applications we get an explicit formula for π₂(C(P _k ,K) ^K ), whereP _k denotes the principal S³-bundle over S⁴ of Chern numberk and derive explicit formulae for the rational homotopy groups π _n (C(P,K) ^K )??.     

Singular gradient flow of the distance function and homotopy equivalence

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ .     

André-Quillen cohomology and rational homotopy of function spaces

We develop a simple theory of André-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent CW-complexes. This puts certain results of Sullivan in a more conceptual framework.