Abstract:
This PhD thesis deals with the topological games theory and the study of the braid group representations. In the first part, we presente many categories of topological games on a topological space X. We show how this games can be used to characterize some topological properties of countability and completeness. given a topological space X, we consider a non empty family γ of compact subsets of X, with this family we provide C(X) ( the set of all real valued continuous functions on X) with a set open topology. This space will be denoted by Cγ(X). After we define two topological games introduce by R.A. McCoy and I. Ntantu which will be used to characterize when Cγ(X) is weakly-α-favorable space. The second part deals with the categorification of the coefficients matrix representation of Burau. We assign to each braid σ, a chain complex such that the graded Euler characteristic of its homology is equal to the coefficient of the Burau matrix. This will allow us to find the faithfulness results given by M. Khovanov and P. Seidel.
