A generalized lower bound theorem for balanced manifolds
  A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $\Delta$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $d-1 \geq 3$, then $2h_2(\Delta) - (d-1)h_1(\Delta) \geq 4{d \choose 2}(\tilde{\beta}_1(\Delta)-\tilde{\beta}_0(\Delta))$, thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $h''$-vectors. 
Martina Juhnke-Kubitzke, Satoshi Murai, Isabella Novik, Connor Sawaske
math.CO math.AC