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

TL;DR

This paper develops the Dax invariant as a powerful tool for understanding the first potentially nontrivial homotopy groups of spaces of embeddings of arcs and circles into $d$‑manifolds with $d\ge3$, linking these groups to immersion spaces and the embedding calculus framework. It provides explicit formulae expressing $\mathsf{Dax}$ and its self‑homotopy variant $\mathsf{dax}$ in terms of the equivariant intersection pairing $\lambda$ and the group ring $\mathbb{Z}[\pi_1X]$, and proves that for $d\ge4$ the Dax map identifies the kernel of the immersion–embedding restriction with a computable abelian group, while in 3‑dimensions it yields universal type $\le1$ Vassiliev invariants for knotted arcs and knots, recovering Schneiderman’s invariant in the appropriate concordance context. The results yield concrete computations in simply connected and aspherical cases, connect to Budney–Gabai’s work, and illuminate how these invariants interact with Wall‑type phenomena and the Goodwillie–Weiss tower. The work thus provides a coherent, computable framework for distinguishing embeddings from immersions at the first nontrivial homotopy level and offers several open questions about the dependence on manifold homotopy type and on the associated central extensions.

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 $M$ of dimension $d\geq4$. In particular, if $M$ is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of $M$, answering a question posed by Arone and Szymik. The case $d=3$ gives isotopy invariants of knots in a 3-manifold, that are universal of Vassiliev type $\leq1$, and reduce to Schneiderman's concordance invariant.

Figures (10)

• Figure 1: Left. The family $\partial\mathfrak{r}(g)(\vec{t})\in\mathop{\mathrm{Emb}}\nolimits_\partial(\mathbb{D}^1,X)$ for several values $\vec{t}\in\mathbb{S}^{d-3}$ and $d=4$ (coloured arcs are in past or future). Right. The single immersed arc $\rho$ in a homotopy from $\partial\mathfrak{r}(g)$ to $\mathsf{const}_\mathsf{u}$ has one double point $x=\rho_{\theta_x^-}=\rho_{\theta_x^+}$, with sign $+1$ and loop $\rho_{\leq\theta_x^-}\cdot \rho_{\leq\theta_x^+}^{-1}\simeq\rho_{\leq\theta_x^-}\cdot \mathsf{u}_{\leq\theta_x^+}^{-1}\simeq g$, so $\mathsf{Dax}(\partial\mathfrak{r}(g))=g$.
• Figure 2: The setting of Theorem \ref{['thm:circles-main']} and Corollary \ref{['cor:gPhi']}. Removing a $d$-ball from a manifold $N$, leaves a boundary component $\Phi$ and turns an embedded circle $\mathsf{s}$ into a neatly embedded arc $\mathsf{u}$. The push-off $\Phi'$ is a $(d-1)$-sphere that intersects $\mathsf{u}$ in two points of opposite sign.
• Figure 3: Left. The $(d-3)$-family of circles $c(g)\coloneqq\partial\mathfrak{r}(g)\cdot\mathsf{s}|_{[e,e+\epsilon]}$ in $N$. Right. After an isotopy of $c(g)$ in $N$, the restriction to $N\mathbin{\mspace{-6mu} \raisebox{\rsmraise{\displaystyle}\depth}{origin=c]{-25}{$\smallsetminus$}} \mspace{-6mu} } D^d$ is a family of arcs with $\mathsf{Dax}(\partial\mathfrak{r}(g)^{new})=(-1)^{d-3}[\widetilde{\rho}_{\leq\theta^-_x}\cdot\widetilde{\rho}_{\leq\theta^+_x}^{-1}]=(-1)^{d-3}\bm\mathsf{s}^{-1}g^{-1}$.
• Figure 4: Left. The double point loops $g_{x_1}=[\rho_{\leq\theta_{x_1}^-}\cdot\rho_{\leq\theta_{x_1}^+}^{-1}]$ and $g_{x_2}=[\rho_{\leq\theta_{x_2}^-}\cdot\rho_{\leq\theta_{x_2}^+}^{-1}]=[\rho_{\leq\theta_{x_2}^-}\cdot\mathsf{u}_{\leq\theta_{x_2}^+}^{-1}]$.
• Figure 5: Left. An immersed arc $F(\vec{t}_i)$ with a double point $x_i=F(\vec{t}_i)(\theta_i^-)=F(\vec{t}_i)(\theta_i^+)$. Right. The red double point loop $g_{x_i}$ is homotopic to $g$ (first follow the dashed arc, then the solid red arc).

Theorems & Definitions (46)

• Theorem 1.1: KT-LBT
• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 1.2
• proof
• Corollary 2.1
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4