Connecting rational homotopy type singularities

Abstract

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$ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $. Here Γ is a a countable union of C¹ curves in R^p+1 with Hausdorff $ {\user1{\mathcal{H}}}^{1} $-measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function θ: Γ→R. The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ 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 $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ 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.