The first homology group of a complete flag complex

The geometry of a flag complex is explored and the homology group $H_1$ computed for the complete flag $F(s)$ by viewing it as a simplicial complex. The homomorphisms between the vertices are extended to the free abelian groups generated by the vertices and it is proved that $H_1(F(s))\cong\underbrace{\mathbb Z+\mathbb Z+\cdots+\mathbb Z}_{k \ \hbox{times}}, \ k\geq1$ where $k=s-r$ , $r$ the rank of a matrix $A$ associated to $F(s)$.