Connected sum of orientable surfaces and Reidemeister torsion

Let $\Sigma_{g,n}$ be an orientable surface with genus g ≥ 2 bordered by $n \geq 1$ curves homeomorphic to circle. As is well known that one-holed torus $\Sigma_{1,1}$ is the building block of such surfaces. By using the notion of symplectic chain complex, homological algebra techniques and considering the double of the building block, the present paper proves a novel formula for computing Reidemeister torsion of one-holed torus. Moreover, applying this result and considering $\Sigma_{g,n}$ as the connected sum $\Sigma_{1,n} \# (g-1) \Sigma_{1,0}$, the present paper establishes a novel formula to compute Reidemeister torsion of $\Sigma_{g,n}$.

Keywords

Reidemeister torsion, symplectic chain complex, homological algebra, orientable surfaces

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This research was supported by TÜBİTAK (project no. 114F516). The first author would also like to thank TÜBİTAK for the financial support.

Received 24 January 2017

Published 26 July 2018