On complex highly regular embeddings and the extended Vassiliev\n conjecture

2014/10/22 by Pavle V. M. Blagojević, Blagojević, Pavle V. M., Fred Cohen +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1410.6052

Abstract

A continuous map C^d -> C^N is a complex k-regular embedding if any k\npairwise distinct points in C^d are mapped by f into k complex linearly\nindependent vectors in C^N. Our central result on complex k-regular embeddings\nextends results of Cohen & Handel (1978), Chisholm (1979) and Blagojevic,\nL "uck & Ziegler (2013) on real k-regular embeddings: We give new lower bounds\nfor the existence of complex k-regular embeddings. These are obtained by\nmodifying the framework of Cohen & Handel (1978) and a study of Chern classes\nof complex regular representations. The main technical result, used for the\nstudy of the Chern classes, is an upper bound for the height of the cohomology\nof an unordered configuration space Furthermore, we give similar lower bounds\nfor the existence of complex l-skew embeddings C^d -> C^N, for which we require\nthat the images of the tangent spaces at any l distinct points are skew complex\naffine subspaces of C^N.\n

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