On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

Abstract

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M M of dimension d ≥ 4 d\geq 4 . In particular, if M M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M M , answering a question posed by Arone and Szymik. The case d = 3 d=3 gives isotopy invariants of knots in a 3-manifold that are universal of Vassiliev type ≤ 1 \leq 1 , and reduce to Schneiderman’s concordance invariant.

Knotted families from graspers

“…the kernel of the surjection 𝜋 𝑑−3 Emb 𝜕 (𝐶, 𝑀) ↠ 𝜋 𝑑−3 Imm(𝐶, 𝑀)) is a quotient of ℤ[𝜋 1 𝑀]. Indeed, the inverse of 𝔯 1 is an explicit map 𝖣𝖺𝗑 coming from the work of Dax [10], as shown in [25], and in [12] for 𝑑 = 4, and in [24] for 𝐶 = 𝕊 1 . Equivalently, and relevantly for this paper, 𝖣𝖺𝗑 = 𝖾𝗏 𝗋𝖾𝗅 2 is the map to the second layer of the Taylor tower constructed by Goodwillie and Weiss [37].…”

“…For 𝑛 = 1, we have 𝑇 1 (𝐶, 𝑀) = Imm(𝐶, 𝑀) and 𝖫𝗂𝖾 𝜋 1 𝑀 (1) = ℤ[𝜋 1 𝑀], and 𝔯 1 is precisely the realisation map from (2), with the inverse 𝖾𝗏 𝗋𝖾𝗅 2 = 𝖣𝖺𝗑. This case is studied in detail in [24], with the goal of making 𝖾𝗏 𝗋𝖾𝗅 2 = 𝖣𝖺𝗑 as computable in practice as possible.…”

“…Indeed, the inverse of $$\mathfrak {r}_1$$ is an explicit map $$\mathsf {Dax}$$ coming from the work of Dax [10], as shown in [25], and in [12] for $$d=4$$, and in [24] for $$C=\mathbb {S}^1$$. Equivalently, and relevantly for this paper, $$\mathsf {Dax}=\mathsf {ev}^{\mathsf {rel}}_2$$ is the map to the second layer of the Taylor tower constructed by Goodwillie and Weiss [37].…”

“…For $$n=1$$, we have $$T_1(C,M)=\operatorname{Imm}(C,M)$$ and $$\mathsf {Lie}_{\pi _1M}(1)=\mathbb {Z}[\pi _1M]$$, and $$\mathfrak {r}_1$$ is precisely the realisation map from (2), with the inverse $$\mathsf {ev}^{\mathsf {rel}}_2=\mathsf {Dax}$$. This case is studied in detail in [24], with the goal of making $$\mathsf {ev}^{\mathsf {rel}}_2=\mathsf {Dax}$$ as computable in practice as possible.…”

Models for knot spaces and Atiyah duality

“…Remark 1. 4 In the recent preprint [23], Kosanović gave a proof of a complete answer to the question, which states that the inclusion i M induces an isomorphism of d 1 if M is simply connected and of dimension d 4 by an independent method.…”