  One of the most interesting entities of homotopy type theory is the identity
type. It gives rise to an interesting interpretation of the equality, since one
can semantically interpret the equality between two terms of the same type as a
collection of homotopical paths between points of the same space. Since this is
only a semantical interpretation, the addition of paths to the syntax of
homotopy type theory has been recently proposed by De Queiroz, Ramos and De
Oliveira . In these works, the authors propose an entity known as
`computational path', proposed by De Queiroz and Gabbay in 1994, and show that
it can be used to formalize the identity type. We have found that it is
possible to use these computational paths as a tool to achieve one central
result of algebraic topology and homotopy type theory: the calculation of
fundamental groups of surfaces. We review the concept of computational paths
and the $LND_{EQ}-TRS$, which is a term rewriting system proposed by De
Oliveira in 1994 to map redundancies between computational paths. We then
proceed to calculate the fundamental group of the circle, cylinder, M{\"o}bius
band, torus and the real projective plane. Moreover, we show that the use of
computational paths make these calculations simple and straightforward, whereas
the same result is much harder to obtain using the traditional
code-encode-decode approach of homotopy type theory.
