The classifying space of a bound quiver

Abstract

We associate to a bound quiver (Q,I) a CW-complex which we denote by B(Q,I), and call the classifying space of (Q,I). We show that the fundamental group of B(Q,I) is isomorphic to the fundamental group of (Q,I). Moreover, we show that this construction behaves well with respect to coverings. On the other hand, we study the (co)homology groups of B(Q,I), and compare them with the simplicial and the Hochschild (co)homology groups of the algebra A=kQ/I. More precisely, we give sufficient conditions for these groups to be isomorphic. This generalizes a theorem due to Gerstenhaber and Schack.