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2008 Connecting rational homotopy type singularities
Robert Hardt, Tristan Rivière
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Acta Math. 200(1): 15-83 (2008). DOI: 10.1007/s11511-008-0023-6


Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W1, p(Rp+1, N) be the Sobolev space of measurable maps from Rp+1 into N whose gradients are in Lp. The restriction of u to almost every p-dimensional sphere S in Rp+1 is in W1, p(S, N) and defines an homotopy class in πp(N) (White 1988). Evaluating a fixed element z of Hom(πp(N), R) on this homotopy class thus gives a real number Φz, u(S). The main result of the paper is that any W1, p-weakly convergent limit u of a sequence of smooth maps in C(Rp+1, N), Φz, u has a rectifiable Poincaré dual$ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $. Here Γ is a a countable union of C1 curves in Rp+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 Φz, u(S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer nz, 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 + nz) 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 Rm for any m ⩾ p + 1, in which case the bubbling is described by an (mp)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.


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Robert Hardt. Tristan Rivière. "Connecting rational homotopy type singularities." Acta Math. 200 (1) 15 - 83, 2008.


Received: 31 January 2005; Revised: 21 May 2007; Published: 2008
First available in Project Euclid: 31 January 2017

zbMATH: 1148.58003
MathSciNet: MR2386108
Digital Object Identifier: 10.1007/s11511-008-0023-6

Rights: 2008 © Institut Mittag-Leffler


Vol.200 • No. 1 • 2008
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