# Equivalent condition of cone

Consider the simplicial complex of independent sets of a matroid. The reduced Euler characteristic of a simplicial complex $\bigtriangleup$ is defined by $\chi(\bigtriangleup)=-1+f_1-f_2+...+(-1)^{r+1}f_r$, where $f_j$ denote the number of faces of cardinality $j$ and $r-1$ is the dimension of $\bigtriangleup$. I want to prove that if Euler characteristic of $\bigtriangleup$ is 0 then $\bigtriangleup$ must be a cone. For me a simplicial complex is cone if all it's maximal faces have a common vertex. This is an exercise of "The homology and Shellability of Matroids and Geometric Lattices" by Bjorner A.

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