On The Simple Connectedness Of A Class Of Incidence Algebras

The importance of representation-finite simply connected algebras has been widely noticed since it was firstly introduced by K.Bongartz and P.Gabriel in 1981/82 (see [1]). In fact, covering techniques allow us to reduce many problems to the problems about simply connected algebras. However, it is well know that currently there are no general methods to decide whether or not a given algebra is simply connected. So it is a meaning full work to decide which kinds of algebras are simply connected and which are not. This thesis deals mainly with a special class of algebras, that is, the incidence algebras whose fundamental group is independent of the presentation. This thesis shows the decomposition of quasi-suspension and it provides convenient criteria to decide the simple connectedness of a kind of incidence algebras. Most of our results are proved by combinatorial approaches.The backgrounds and our main results are presented in the first chapter. At the second chapter, we extend the conceptions of suspend and suspension to the cases of quasi-suspend and quasi-suspension respectively. In this chapter, we also explore the nature of crown from different perspectives which make us more convenient to check whether or not a given cycle is a crown. This view helps us to identify the crown as a cycle with some special properties. In the third chapter, we begin to attack our first main result: the decomposition of the quasi-suspension. This decomposition is meaningful since it reduces the exploration of the structure of crownΓin incidence algebras to a relative more accessible case due to the Proposition 3.2.3 (b) since the circumference of the weak crown is homotopic to a conjugate of the products of circumferences, all starting and ending at the common quasi-suspend point, of the crown in the decomposition. This proposition also indicates the relationship of the widths between the crowns in the decomposition and the original crown. To help understand the proposition, an example was presented in advance. At last, in chapter 4, lemma 4.1.3 provides us a procedure to check whether or not a given cycle is crown. Furthermore, theorem 4.2.1. gives us a convenient criterion to judge a large amount of incidence algebras that are not simply connected. In addition, corollary 4.2.2. shows a case that when the quasi-suspension is not simply connected.