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Yayın Gluing formulas for volume forms on representation varieties of surfaces(Springer Nature, 2025-08-06) Erdal, Esma DiricanLet Σg,n be a compact oriented surface of genus g≥4 with n boundary components. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah–Bott–Goldman–Narasimhan symplectic form on the space of representations of π1(Σg,0) in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of Σg,0 in terms of torsions of Σg1,1,Σg2,1, and boundary circle S1, where g=g1+g2 and g1,g2≥2. Then, by using Heusener and Porti’s results on Σg,n, we show that the symplectic volume form on the representation variety of Σg,0 can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces Σg1,1 and Σg2,1.Yayın Higher analogues of discrete topological complexity(Springer-Verlag Italia S.R.L., 2024-06-13) Alabay, Hilal; Borat, Ayşe; Cihangirli, Esra; Erdal, Esma DiricanIn this paper, we introduce the nth discrete topological complexity and study its properties such as its relation with simplicial Lusternik–Schnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n-th discrete topological complexity which is given by the nth usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.Yayın Higher analogues of discrete topological complexity(Cornell Univ, 2024-04-16) Alabay, Hilal; Borat, Ayşe; Cihangirli, Esra; Erdal, Esma DiricanIn this paper, we introduce the n−th discrete topological complexity and study its properties such as its relation with simplicial LusternikSchnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n−discrete topological complexity which is given by the n−th usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.Yayın On twisted torsion of compact 3-manifolds(Cornell Univ, 2024-08-20) Erdal, Esma DiricanLet M be a 3-manifold with connected non-vacuos boundary which is not spherical. Assume that N is another 3-manifold with vacuous boundary and N∗ is the 3-manifold obtained by removing from N the interior of a 3-cell. In the present paper, we find a relationship between the multiplicative property of the twisted Reidemeister torsion and the connected sum operation on these manifolds in order to understand their topology and geometry.
