semi-fredholmness on a weighted geometric realization of 2-simplexes and 3- simplexes

In this present article, we introduce the notion of oriented $2$-simplexes and the notion of oriented $3$-simplexes and we use them to create a new framework that we call a weighted geometric realization of $2$-simplexes and $3$-simplexes. next, we define the weighted geometric realization gauss-bonnet operator $l$. after that, we present and study the non-parabolicity at the infinity of  $l$. finally, we develop general conditions to ensure semi-fredholmness of $l$ based on its non-parabolicity at infinity.