Generalized plumbings and Murasugi sums

TL;DR

This paper generalizes classical plumbing and Murasugi sum operations to higher-dimensional smooth manifolds, establishing their compatibility with open book decompositions and Morse maps, and explores implications for singularity theory and contact topology.

Contribution

It introduces a broad generalization of plumbing and Murasugi sums applicable to arbitrary dimensions, maintaining their geometric significance and linking to open book structures.

Findings

Sum of pages of two open books is again a page of an open book

Established an associated summing operation for Morse maps

Connected the generalized sums to singularity theory and contact topology

Abstract

We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory or contact topology.