Bott-Taubes-Vassiliev cohomology classes by cut-and-paste topology
Abstract
Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in Euclidean spaces of dimension greater than three. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-p classes out of mod-p graph cocycles, which need not be reductions of classes over the integers.
Keywords
Cite
@article{arxiv.1512.06654,
title = {Bott-Taubes-Vassiliev cohomology classes by cut-and-paste topology},
author = {Robin Koytcheff},
journal= {arXiv preprint arXiv:1512.06654},
year = {2021}
}
Comments
Most significant revisions from v4: corrected Main Theorem statement by an integer factor; expanded material on pushforward and its relation to fiber integration for glued manifolds; replaced cellular treatment of fundamental class by homological treatment; and added material on blowups, neatness, and mutual transversality. 48 pages, 4+ figures. Accepted for publication in Internat. J. Math