A space level light bulb theorem in all dimensions

TL;DR

The authors establish a space-level version of the light bulb trick for disk embeddings in arbitrary dimensions with a boundary dual, showing $\mathrm{Emb}_s(\mathbb{D}^k,M)\simeq \Omega\mathrm{Emb}_{u_0}^\varepsilon(\mathbb{D}^{k-1},M_G)$ after attaching a handle along the dual to form $M_G$. The approach combines Cerf’s half-disk trick with ambient isotopy extension to convert neat disks into half-disks in a handle-attached manifold, enabling a delooping via loop-space descriptions. Central to the analysis is Dax’s invariant for relative immersion-embedding groups, which the authors adapt to the simply connected and augmented settings, yielding explicit bordism groups $\mathbb{Z}[\pi_1X]/\mathsf{rel}_{\ell,d}$ and a realizable inverse $\mathfrak{r}$ of $\mathsf{Dax}$; these provide concrete computations of low-degree homotopy groups and exact sequences governing embeddings. The work culminates in a unified framework that, among other things, supplies a complete isotopy classification for 2-disks in 4-manifolds with a dual boundary, and sets the stage for further refinements via embedding calculus and augmented-augmentation phenomena in high dimensions.

Abstract

Given a $d$-dimensional manifold $M$ and a knotted sphere $s\colon\mathbb{S}^{k-1}\hookrightarrow\partial M$ with $1\leq k\leq d$, for which there exists a framed dual sphere $G\colon\mathbb{S}^{d-k}\hookrightarrow\partial M$, we show that the space of neat embeddings $\mathbb{D}^k\hookrightarrow M$ with boundary $s$ can be delooped by the space of neatly embedded $(k-1)$-disks, with a normal vector field, in the $d$-manifold obtained from $M$ by attaching a handle to $G$. This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree $d-2k$. In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

Figures (12)

• Figure 1.1: Correspondence of neat $k$-disks in $M$ with boundary $s$ and half-disks in $M_G$ for $k=1,2$, $d=3$.
• Figure 2.1: Moments $t=0$, $0<t<1$, $t=1$ of an ambient isotopy $\Phi_t$ defining $\mathfrak{a}_\mathrm{U}(\gamma)$. A neighborhood of the intersection $\mathrm{U}\cap\gamma$ has to be "dragged along" all the way during the isotopy, whereas a neighborhood of the undercrossing in $\gamma$ is dragged along only for a while.
• Figure 2.2: Samples $\mathfrak{r}(g)_t\in\mathop{\mathrm{Emb}}\nolimits_\partial(\mathbb{D}^\ell,X)$ for several $t\in \mathbb{S}^{1}$ and $\ell=1,d=4$.
• Figure 3.1: A half-disk for $k=2$, $d=3$.
• Figure 3.2: The image of $\varphi_t$ for $t=2/3,1/3,\varepsilon$. Dashed strips show where $\varphi_t$ is the identity; they are always contained in the blue-colored strip $\mathbb{D}_-^\varepsilon\subseteq\text{\reflectbox{$\mathsf{D}$}}^k$. The black line is the image of $\mathbb{D}^{k-1}\times\{\varepsilon\}\subseteq\mathbb{D}_+^\varepsilon\subseteq\text{\reflectbox{$\mathsf{D}$}}^k$.

Theorems & Definitions (95)

• Theorem 1
• Theorem 2: Space level light bulb theorem for disks
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 3
• Theorem 4
• Remark 1.7
• Remark 1.8
• Lemma 2.2