Tree homology and a conjecture of Levine
| Autor: | James Conant, Peter Teichner, Rob Schneiderman |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: |
medicine.medical_specialty
Whitney tower Discrete Morse theory Homology (mathematics) 01 natural sciences Mathematics::Algebraic Topology Combinatorics quasi-Lie algebra 010104 statistics & probability Mathematics - Geometric Topology 57N10 Mathematics::K-Theory and Homology FOS: Mathematics medicine 0101 mathematics discrete Morse theory Mathematics Intersection theory Conjecture 010102 general mathematics homology cylinder Levine conjecture Geometric Topology (math.GT) Mathematics::Geometric Topology Graph Mapping class group tree homology 57M27 57M25 Geometry and Topology Group homomorphism 57M27 55U15 Singular homology |
| Zdroj: | Geom. Topol. 16, no. 1 (2012), 555-600 Geometry & Topology |
| Popis: | In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism [math] is an isomorphism. Both [math] and [math] are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of [math] . In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain [math] and range [math] of Levine’s map. ¶ The isomorphism [math] is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders. |
| Databáze: | OpenAIRE |
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