Lower bound theorems for manifolds and balanced manifolds

In this talk we will concentrate on finite simplicial complexes (that is, points, line segments, triangles, and higher-dimensional simplices nicely glued together) that triangulate manifolds. A $(d-1)$-dimensional complex is called balanced if its graph is $d$-colorable in the usual graph-theoretic sense. After reviewing what is known about the face numbers of triangulated manifolds without the balancedness assumption, we will discuss several very recent balanced analogs of these results. One of them is a lower bound on the number of edges of a balanced triangulation of a manifold $M$ in terms of the number of vertices and the 1st homology of $M$. The most recent results are joint work with Martina Juhnke-Kubitzke, Satoshi Murai, and Connor Sawaske.

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