Representation varieties of free or surface group and Reidemeister torsion

In this thesis, by using the algebraic topological instrument symplectic chain complex and the toplogical invariant Reidemeister torsion, which has also many applications in several branches of mathematics and theoretical physics, several G-valued representation varieties for free and surface groups are investigated. Let Σ be a closed oriantable surface of genus at least 2 and let G denote one of the matrix groups GL(n;C), SL(n;C), O(2n+1;C), O(2n;C), Sp(2n;C), SO*(2n), U(n), U(p;q), O(p;q) or one of the exceptional groups G_2, F_4, E_6. It is proved that for homomorphisms ρ:π_1(Σ)→G from fundamental group π_1(Σ) to Lie group G, the concept of Reidemeiter torsion is well defined. Moreover, by using symplectic complex method, the Reidemeister torsion of such representations is expressed in terms of the well-known Atiyah-Bott-Goldman symplectic form for the Lie group G. In addition, the obtained results are applied to the representation varieties of the compact 3-manifolds with boundary consisting of closed orientable surfaces with genus at least 2.