An alternative approach to the calculation of fundamental groups based on labeled natural deduction

Abstract

In this work, we use a labelled deduction system based on the concept of computational paths (sequence of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequence of rewrites) to obtain some results of algebraic topology and with support of the Seifet-Van Kampen Theorem we will calculate, in a way less complex than the one made in mathematics \cite{Munkres} and the technique of homotopy type theory \cite{hott}, the fundamental group of Klein Blottle ${\mathbb{K}}^2$, of the Torus ${\mathbb{T}}^2$ and Two holed Torus ${\mathbb{M}}_2={\mathbb{T}}^2\#{\mathbb{T}}^2$ (the connected sum two torus).

28 pages, 17 figures arXiv admin note: text overlap with arXiv:1804.01413, arXiv:1803.01709, arXiv:1906.09105