On simplicial resolutions of framed links

Abstract

Abstract. In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3-spheres. This gives simplicial group models for some loop spaces using link groups.

On degree of simplicial homomorphisms between link groups

“…From the construction of d i and s i , it is routine to show that the simplicial identities hold and so G(L; X) is a simplicial group. The main result in [7] showed that the geometric realization of G(L; X) is homotopy equivalent to the loop space of a wedge of 3-spheres for any non-trivial framed link L.…”

“…( [7]) Suppose L f is a framed link in S 3 with L a trivial knot and the associated naive cabling L n the Hopf link with (n + 1)-components, then the link group G(L; X) = {G(L n )} n 0 is a simplicial group. Furthermore, the geometric realization of G(L; X) is homotopy equivalent to ΩS 3 .…”

“…Proof. Note that from [7], |G(L; X)| is homotopy equivalent to ΩS 3 , hence BG(L; X) S 3 . By Proposition 3.2,F is homotopy equivalent to ΩS 2 , hence BF S 2 .…”

“…(The construction will be briefly reviewed below.) The main result in [7] showed that such the geometric realization of G(L; X) is homotopy equivalent to the loop space of a wedge of 3-spheres for any non-trivial framed link L. The purpose of this article is to explore some properties of these simplicial groups. We aim to the difference between these simplicial groups and Kan's free simplicial group construction GS 3 , which is a classical simplicial group model for ΩS 3 .…”

“…( [7]) Let L f be a framed link in S 3 with the associated twisted-annulus link (L; X, A) and the canonical splitting decomposition (L; X, A) ∼ = L [1] ; X [1] , A [1] L [2] ; X [2] , A [2] · · · L [p] ; X [p] , A [p] such that (L [1] ; X [1] , A [1] After then Prof. Jie Wu asked me that whether the simplicial group model we constructed for ΩS 3 has the same properties as other constructions for ΩS 3 , such as Kan's or Milnor's [8]. In the present paper, we partially answer this question from the perspective of the degree of simplicial homomorphism ϕ: G(L; X) −→ G(L; X).…”