A note on some high-dimensional handlebodies

TL;DR

This work proves a fundamental product-structure for high-dimensional handlebodies: whenever $k\ge0$ and $n\ge 2k+1$, every $n$-dimensional $k$-handlebody $X_k$ is diffeomorphic to a product $Y_k\times B^{n-2k}$ with a $2k$-dimensional $k$-handlebody $Y_k$, and highlights that different $Y_k$ can yield the same product after taking a further ball factor. The central technical advance is an inductive argument that the attaching map of each new $k$-handle can be isotoped to a product form, supported by a sequence of lemmas (notably a Whitney-type isotopy result and a compatibility of framings via induced maps on orthogonal groups). Building on this, the paper introduces $(n,k)$-Kirby diagrams, a natural generalization of classical Kirby diagrams to higher dimensions, and shows how these diagrams classify $n$-dimensional $k$-handlebodies up to diffeomorphism via diagrammatic moves (including crossing changes and $\ddagger$-moves). The construction yields practical tools for translating high-dimensional handlebody questions into (induced) $(4,2)$-diagrams and provides open-book decompositions on boundaries for $n\ge 2k+2$, along with examples and a structural classification involving bundles over spheres. The results illuminate how high-dimensional topology can often be reduced to tractable diagrammatic data while preserving smooth structure, with implications for understanding boundary phenomena and diffeomorphism classification.

Abstract

For $k \geq 0$ and $n \geq 2k+1$, we show that every $n$-dimensional $k$-handlebody is the product of a $2k$-dimensional $k$-handlebody and the standard $(n-2k)$-ball. For $k \geq 2$ and $n \geq 2k$, we introduce $(n,k)$-Kirby diagrams for some $n$-dimensional $k$-handlebodies, where $(4,2)$-Kirby diagrams correspond to the original Kirby diagrams for $4$-dimensional $2$-handlebodies.

Figures (6)

• Figure 1: First row: A crossing change from $L_2$ to $L_2'$ induces an isotopy from the $(k-1)$-link $\tilde{L_2}$ to $\tilde{L_2'}$ in $\mathbb{R}^{n-k-2}\times\mathbb{R}^3\times\mathbb{R}^{k-2}\setminus \tilde{L_1}$, where $k\geq2$. Second row: A $\ddagger$-move from $L_2$ to $L_2'$ induces an isotopy from the $(k-1)$-link $\tilde{L_2}$ to $\tilde{L_2'}$ in $\mathbb{R}^{k-2}\times\mathbb{R}^3\times\mathbb{R}^{n-k-2}$. Note that two parallel strands with opposite orientations belong to the same component of $L_2$. Third row: $L_2$ is obtained from $L_2"$ by surgery along a $2$-dimensional $1$-handle $h_C$, whose core is $C$. Similarly, $L_2'$ is obtained from $L_2"$ by surgery along a $2$-dimensional $1$-handle $h_D$, whose core is $D$.
• Figure 2: Some $(n,k)$-Kirby diagrams.
• Figure 3: Moves defined on $(n,k)$-Kirby diagrams, where $k\geq2$ and $n\geq2k+1$. A dotted circle in $\{0\}\times\mathbb{R}^3\times\{0\}$ represents a dotted trivial $(n-k-1)$-sphere in $\mathbb{R}^{n-k-2}\times\mathbb{R}^3\times\mathbb{R}^{k-2}$. A circle with a number $t$ in $\{0\}\times\mathbb{R}^3\times\{0\}$ represent a $t$-framed trivial $(k-1)$-sphere in $\mathbb{R}^{n-k-2}\times\mathbb{R}^3\times\mathbb{R}^{k-2}$, where $t\in G$. First row: A $(k-1)$-handle slide of $A$ over $B$ is a dotted $(n-k-1)$-sphere $A_{new}=A\#_hB'$ obtained from $A\cup B'$ by surgery along an $(n-k)$-dimensional $1$-handle $h=B^1\times B^{n-k-1}$ attached to $A$ and a parallel copy $B'$ of $B$. Here, we can always isotope the core $B^1\times\{0\}$ of the $1$-handle $B^1\times B^{n-k-1}$ into $\{0\}\times\mathbb{R}^3\times\{0\}\subset \mathbb{R}^{n-k-2}\times\mathbb{R}^3\times\mathbb{R}^{k-2}$, because homotopy implies isotopy for $1$-manifolds in $\mathbb{R}^{n-1}$. In simple terms, $A_{new}$ is obtained from $A\cup B'$ by surgery along a visible $2$-dimensional band $B^1\times B^1\times\{0\}$ of the $1$-handle $B^1\times B^1\times B^{n-k-2}=B^1\times B^{n-k-1}$. Second row: A $k$-handle slide of $t_1$-framed $C$ over $t_2$-framed $D$ is a $(t_1+t_2)$-framed $(k-1)$-sphere $C_{new}=C\#_hD'$ obtained from $C\cup D'$ by surgery along a $k$-dimensional $1$-handle $h=B^1\times B^{k-1}$ attached to $C$ and a parallel copy $D'$ of $D$. Note that there is no linking between a framed $(k-1)$-sphere $D$ and its parallel copy $D'$ when $k\geq2$ and $n\geq2k+1$. Similarly, the core of the $1$-handle can be isotoped into $\{0\}\times\mathbb{R}^3\times\{0\}\subset \mathbb{R}^{n-k-2}\times\mathbb{R}^3\times\mathbb{R}^{k-2}$, and $C_{new}$ is obtained from $C\cup D'$ by surgery along a visible band $B^1\times B^1\times\{0\}$ of the $1$-handle $B^1\times B^1\times B^{k-2}=B^1\times B^{k-1}$. Third row: A cancelling $(k-1,k)$-pair is the union of a dotted $(n-k-1)$-sphere $E$ and a framed $(k-1)$-sphere $F$, with the geometric intersection number is $|D_{E}\pitchfork F|=1$, where $D_{E}$ is the trivial $(n-k)$-ball of $E$ and $E\cap (\{0\}\times\mathbb{R}^3\times\{0\})$ is a visible $2$-disk bounded by the dotted circle.
• Figure 4: $(n,k)$-Kirby diagrams $\mathcal{K}$ and $\mathcal{K}(t)$.
• Figure 5: $(n,k)$-Kirby diagrams $\mathcal{K},\mathcal{K}'$, and $\mathcal{K}(p;a,b)$. Left: The diagram $\mathcal{K}=L_1\cup L_2\subset S^{n-1},$ where $L_1$ is a dotted $(n-k-1)$-sphere, and $L_2=J^{t_1}_{1}\cup \cdots \cup J^{t_m}_m$ is a framed $(k-1)$-link consisting of $t_i$-framed $(k-1)$-knots $J^{t_i}_i$. The components $L_1$ and $L_2$ are linked, while $L_2$ itself forms the trivial $(k-1)$-link. Each $p_i$ denotes $p_i$-full twist. Middle: The modified diagram $\mathcal{K}'=L_1\cup L_2'\subset S^{n-1},$ where $L_2'=N^{a}_{1}\cup N^{s_2}_2\cup \cdots \cup N^{s_m}_m$ is a framed $(k-1)$-link with $N^{s_i}_i$ being an $s_i$-framed $(k-1)$-knot and $N^{a}_1$ an $a$-framed $(k-1)$-knot. The components $L_1$ and $N^{a}_1$ are linked. Here, $p$ presents a $p$-full twist. Right: The simplified diagram $\mathcal{K}(p;a,b)=L_1\cup L_2"$, where $L_2"=N^{a}_{1}\cup E^{b}_2\cup E^{u_3}_3\cup \cdots \cup E^{u_m}_m$ is a framed $(k-1)$-link, with $u_3=\cdots=u_m=0$.

Theorems & Definitions (58)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2
• Corollary 1.3
• proof
• Corollary 1.4
• proof
• Remark 1.5
• Conjecture 1.6