Algebraic fundamental group and simplicial complexes

In this paper we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications. AMS classification : 16E40 ; 16G20 ; 06A11 ; 55Q05 Let k be a field and A be a basic and split finite dimensional k-algebra, which means that A/r = k× k× . . .× k where r is the radical of A. There exists a unique quiver Q and usually several admissible ideals I of the algebra kQ such that A = kQ/I (see [6]). In the 1980s, an algebraic fundamental group has been defined which depends on the presentation of A, that is to say on the choice of the ideal I (see [13]). For incidence algebras, that is algebras obtained from a simplicial complex, it has been proved that the presentation does not influence the fundamental group ([15]). Then it is a natural question to compare it with the fundamental group of the geometric realisation. Note also that in [4,8] the analogous question concerning homology is solved. Actually, we prove that the fundamental groups considered for a finite and connected simplicial complex are isomorphic. The following diagram summarizes the situation : Simplicial complex Geometric realization Topological fundamental group Poset Incidence algebra Algebraic fundamental group 1