The topology of the moment-angle manifolds--On a conjecture of S. Gitler ans S. Lopez

Let $P$ be a simple polytope of dimension $n$ with $m$ facets and $P_{v}$ be a polytope obtained from $P$ by cutting off one vertex $v$. Let $Z=Z(P)$ and $Z_{v}=Z(P_{v})$ be the corresponding moment-angle manifolds. In \cite{[GL]} S.Gitler and S.L\'{o}pez conjectured that: $Z_{v}$ is diffeomorphic to $\partial[(Z-int(D^{n+m}))\times D^{2}]\sharp \mathop{\sharp} \limits_{j=1}^{m-n} \binom{m-n}{j} (S^{j+2}\times S^{m+n-j-1})$, and they have proved the conjecture in the case $m<3n$. In this paper we prove the conjecture in general case.