Connecting rational homotopy type singularities |
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Authors: | Robert Hardt Tristan Rivière |
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Affiliation: | (1) Department of Mathematics, Rice University, P.O. Box 1892, Houston, TX 77251, USA;(2) Department of Mathematics, Swiss Federal Institute of Technology, CH-8092 Zürich, Switzerland |
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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 . 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. |
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