On LS-category and topological complexity of connected sum

TL;DR

This paper investigates how the Lusternik-Schnirelmann category and topological complexity invariants behave under connected sum operations on manifolds, providing exact formulas and inequalities for these invariants.

Contribution

It establishes that the LS-category of orientable manifolds under connected sum equals the maximum of the individual categories, and proves a lower bound for topological complexity.

Findings

LS-category of orientable manifolds under connected sum equals the maximum of the individual categories

Topological complexity of connected sums is at least the maximum of the individual complexities for simply connected manifolds

Provides complete characterization for LS-category and bounds for topological complexity in connected sums

Abstract

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$. For topological complexity we prove the inequality $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.