A note on some high-dimensional handlebodies

TL;DR

This paper demonstrates a decomposition of high-dimensional handlebodies into products of lower-dimensional handlebodies and balls, and introduces a new diagrammatic representation for these structures, extending Kirby diagrams to higher dimensions.

Contribution

It provides a new decomposition theorem for high-dimensional handlebodies and introduces $(n,k)$-Kirby diagrams, generalizing existing diagrams to higher dimensions.

Findings

Every $n$-dimensional $k$-handlebody decomposes into a product with a $2k$-dimensional handlebody and a ball.

Introduction of $(n,k)$-Kirby diagrams for visualizing high-dimensional handlebodies.

$(4,2)$-Kirby diagrams recover the classical Kirby diagrams for 4-dimensional 2-handlebodies.

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.