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GRADED POISSON ALGEBRAS ON BORDISM GROUPS OF GARLANDS AND THEIR APPLICATIONS (2006)
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BibTeX
@MISC{Chernov06gradedpoisson,
author = {Vladimir V. Chernov},
title = {GRADED POISSON ALGEBRAS ON BORDISM GROUPS OF GARLANDS AND THEIR APPLICATIONS},
year = {2006}
}
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Abstract
Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of N-garlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our previous work with Rudyak we introduced a rich algebra structure on the bordism group Ω∗(GN,M). In this work we introduce the operations ⋆ and [·, ·] on the tensor product of Q and a certain bordism group ̂Ω∗(GN,M). For N consisting of odd-dimensional manifolds, these operations make ̂Ω∗(GN,M) ⊗ Q into a graded Poisson algebra (Gerstenhaber-like algebra). For N consisting of even-dimensional manifolds, ⋆ satisfies a graded Leibniz rule with respect to [·, ·], but [·, ·] does not satisfy a graded Jacobi identity. The mod 2-analogue of [·, ·] for one-element sets N was previously constructed in our preprint with Rudyak. For N = {S 1} and a surface F 2, the subalgebra ̂ Ω0(G {S 1},F 2) ⊗ Q of our algebra is related to the Goldman-Turaev algebra of loops on a surface and to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes ̂ δ1, ̂ δ2 of free loops on F 2, provided that ̂ δ1, ̂ δ2 do not contain powers of the same element of π1(F 2).
Keyphrases
free loop gerstenhaber-like algebra rich algebra structure goldman-turaev algebra intersection point one-element set previous work graded poisson algebra even-dimensional manifold singular manifold odd-dimensional manifold lie bracket andersen-mattes-reshetikhin poisson algebra graded jacobi identity marked point space gn set consisting graded leibniz rule different dimension bordism group closed manifold oriented manifold certain bordism group tensor product minimal number
