A homotopy-theoretic view of Bott-Taubes integrals and knot spaces (Q834381)

Let \(\text{Emb}(S^1, \mathbb{R}^n)\) be the space of embeddings of the circle in \(\mathbb{R}^n\). When \(n=3\), this is the space of classical knots. In [J. Math. Phys. 35, No.~10, 5247--5287 (1994; Zbl 0863.57004)], \textit{R. Bott} and \textit{C. Taubes}, and then, following their model, \textit{A. S. Cattaneo, P. Cotta-Ramusino} and \textit{R. Longoni} [Algebr. Geom. Topol. 2, 949--1000 (2002; Zbl 1029.57009)] constructed certain cohomology classes on \(\text{Emb}(S^1, \mathbb{R}^n)\) as follows: They first constructed a bundle \(E_{q,t}\) over \(\text{Emb}(S^1, \mathbb{R}^n)\) whose fiber over a knot \(K\) is the compactified configuration space of \(q+t\) points in \(\mathbb{R}^n\), \(q\) of which must lie on \(K\). A suitable differential form is then integrated along this bundle. Lastly, they showed that certain linear combinations of such integrals produce cohomology classes on \(\text{Emb}(S^1, \mathbb{R}^n)\). The main result of the paper under review is the replacement of the integration along the fiber in the above construction by a more homotopy-theoretic way of ``pushing forward'' the forms on \(E_{q,t}\) to \(\text{Emb}(S^1, \mathbb{R}^n)\). The author does this via a Thom collapse map which essentially sends the \(N\)-fold suspension of \(\text{Emb}(S^1, \mathbb{R}^n)\) to the normal bundle, modulo boundary, of \(E_{q,t}\) embedded in \(\mathbb{R}^N\) (via a \textit{neat} embedding; this is an important technical detail). These maps are compatible in \(N\) so that one gets a map from the suspension spectrum of \(\text{Emb}(S^1, \mathbb{R}^n)\) to the corresponding Thom spectrum. The author also defines a multiplication on \(E_{q,t}\) modulo its boundary which is compatible with the operation of connected sum of knots, and this multiplication gives the wedge of Thom spectra, over various \(q\) and \(t\), a ring spectrum structure. The map induced on cohomology by a map from this wedge to a wedge of spaces \(E_{q,t}\) modulo boundary is what is ultimately called the homotopy-theoretic analog of the Bott-Taubes integration. One goal of the paper is evaluation of the resulting cohomology classes on the Budney-Cohen prime homology classes [\textit{R. Budney} and \textit{F. Cohen}, Geom. Topol. 13, No.~1, 99--139 (2009; Zbl 1163.57027)]. To this end, a product formula is also given, but evaluation on the Browder operation coming from Budney's action of the little discs operad on \(\text{Emb}(S^1, \mathbb{R}^n)\) still remains to be examined (or on Dyer-Lashof operations in mod \(p\) homology). An advantage of this construction is that \textit{integral} cohomology classes on \(\text{Emb}(S^1, \mathbb{R}^n)\) are produced, while the original Bott-Taubes integration only produced real classes. It would be nice to examine how closely related the two constructions really are. In particular, it would be nice to see if, when \(n=3\), the construction in this paper produces finite type invariants, as the original Bott-Taubes integration does.