# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.331990

Acta Mathematica, 200, (1), 15-83, 2005.1
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$C$^{1}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.
@inproceedings{robert2005connecting,
title={Connecting rational homotopy type singularities},
author={Robert Hardt, and Tristan Rivière},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351466016699},
booktitle={Acta Mathematica},
volume={200},
number={1},
pages={15-83},
year={2005},
}

Robert Hardt, and Tristan Rivière. Connecting rational homotopy type singularities. 2005. Vol. 200. In Acta Mathematica. pp.15-83. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351466016699.