Goodwillie's cosimplicial model for the space of long knots and its applications

TL;DR

This paper explores Goodwillie's cosimplicial model for long knots, providing detailed computations of the associated spectral sequence and introducing combinatorial tools for understanding differentials.

Contribution

It develops a detailed correspondence between cosimplicial spaces and good functors, and offers a combinatorial interpretation of spectral sequence differentials for long knots.

Findings

Computed the first page of the Bousfield--Kan spectral sequence for long knots.

Introduced $(i, n)$-marked unitrivalent graphs to interpret differentials.

Connected cosimplicial models with combinatorial graph structures.

Abstract

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield--Kan homotopy spectral sequence of the tower of fibrations given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in [Con08], we give a combinatorial interpretation of the differentials $d^1$ mapping into the diagonal terms, by introducing the notion of $(i, n)$-marked unitrivalent graphs.