Operads and embeddings

In this thesis, our objective is to present a strategy of a new proof of the weak equivalence Emb(R1, Rm) ∼ Ω2MapOp(D1,Dm), where Emb(R1, Rm) is the space of tangentially straightened long knots in Rm (see [1]) and MapOp(D1,Dm) is the space of operadic morphisms from the little 1-disk operad to the little m-disk operad. The existing proofs of Turchin [2] and Dwyer-Hess [1] are based on homotomy theory. We develop a more categorical proof which uses the theory of internal algebra classifiers [3] and explains conceptually the 'raison d'etre' of such a delooping. It also allows us to employ powerfu categorical/combinatorial techniques developed in [3] for proving and generalising of this sort of results. Our proof should admit a generalisation to higher dimensions, known as Dwyer-Hess conjecture.