Natural central extensions of groups

Abstract

Given a group $G$ and an integer $n\ge 2$, we construct a new group $\tilde{\mathcal{K}}(G,n)$. Although this construction naturally occurs in the context of finding new invariants for complex algebraic surfaces, it is related to the theory of central extensions and the Schur multiplier. A surprising application is that Abelian groups of odd order possess naturally defined covers that can be computed from a given cover by a kind of warped Baer sum.