Some fundamentals for Nielsen theory on torus configuration spaces

The braid groups and configuration spaces of Euclidean space is a well documented. Other braid spaces have been used in papers to prove or disprove. One classic example of this is in [Jiang] where Boju Jiang uses the pure 2-braid group on the so called pants space to give an example of a space with negative Euler characteristic which does not have the Wecken property. We study the pure (or colored) braid groups on the 2-torus and present a geometric interpretation of the group which we then use to calculate a presentation of the pure 2-braid group as an extension of the fundamental group of the 2-torus. To accomplish this, we use the Fadell-Neuwirth exact sequence as well as an algebraic result to show that the pure 2-braid group of the torus is isomorphic to an algebraic group similar to but slightly different than the semi direct product of two free groups of rank 2. Using a method inspired by Jiang, we calculate the relations for the 2-braid group on the 2-torus. We then use the algebraic structure of this braid group to induce endomorphisms on the braid group which can be seen as being induced by self maps on the second configuration space via a fibration structure. From this we can then show that the maps are fiber-preserving and satisfy the naive addition constraints of Heath-Keppelmann-Wong, which allows us to compute the Nielsen Fixed point number of maps on the second configuration space. This gives an interesting example of a fiber space different from the solvmanifold which satisfies the naive addition formula.