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     

On the rational homotopy type of function spaces

The main result of this paper is the construction of a minimal model for the function space of continuous functions from a finite type, finite dimensional space to a finite type, nilpotent space in terms of minimal models for and . For the component containing the constant map, in positive dimensions. When is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for and the coproduct of . We also give a version of the main result for the space of cross sections of a fibration.

     

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.     

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if lies in the class and is a torus of positive dimension, then ``most' vector bundles over admit no complete nonnegatively curved metrics.

     

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.     

Examples of rational homotopy types of blow-ups

We give an example of two homotopic embeddings of manifolds with isomorphic complex normal bundles but such that the blow-ups of along and along have different rational homotopy types.

     

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 )??.     

Conformal cochains

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author's paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.