Retraction of the complement of smooth projective curves to a $2$-dimensional $Δ$-complex

Abstract

Due to a result by Andreotti and Frankel \cite{andreotti1959}, it can be seen that the complement of a complex projective curve has the homotopy type of a $2$-dimensional CW complex. However, no general method has been given to compute explicitly this complex. Here we give a explicit construction of a $2$- dimensional $Δ$-complex that is a strong deformation retract of the complement of a Fermat curve of degree $d$ in the complex projective space. The retraction is performed in several steps, using the branched cover structure of the Fermat curves over the degree $1$ case.