On LS-category and topological complexity of connected sum

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, $\text{cat}(M\mathrm{\#}N)=\max \{\text{cat}M,\text{cat}N\}$. For topological complexity we prove the inequality $\text{TC}(M\mathrm{\#}N)\ge \max \{\text{TC}M,\text{TC}N\}$ for simply connected manifolds.