Given a $4d$ simplicial complex (a triangulation of $4$-manifold), is there any relation between the number of $2$-simplices (triangles) and the number of $1$-simplices (edges)? Generically, is the number of $2$-simplices always greater than the number of $1$-simplices? Thanks!
2026-03-26 19:02:54.1774551774
The number of $2$-simplices and the number of $1$-simplices in a $4$-dimensional simplicial complex
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The answer to the generic question is a negative. Consider the 4-simplex, it has ten 1-simplices and ten 2-simplices.