Bott-Taubes/Vassiliev cohomology classes by cut-and-paste topology

Robin Koytcheff

Published 2015-12-21Version 1

Bott and Taubes used integrals over configuration spaces to produce finite-type (a.k.a. Vassiliev) knot invariants. Their techniques were then used to construct "Vassiliev classes" in the real cohomology spaces of knots and links in higher-dimensional Euclidean spaces, using classes in graph cohomology. Here we construct integer-valued cohomology classes in spaces of knots and links in odd-dimensional 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 constructing mod-p classes out of mod-p graph cocycles, which need not be reductions of classes over the integers.