A necessary condition for the tightness of odd-dimensional combinatorial manifolds

We present a necessary condition for (ℓ−1)-connected combinatorial (2ℓ+1)-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with first Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Klein bottle.