Stable integral simplicial volume of 3‐manifolds

Stable complexity and simplicial volume of manifolds

Let the Δ-complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M . Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree, we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, whi...

Integral Homology of Random Simplicial Complexes

`1-Homology and Simplicial Volume

Introduction A pervasive theme of contemporary mathematics is to explore rigidity phenomena caused by the symbiosis of algebraic topology and Riemannian geometry on manifolds. In this context, the term " rigidity " refers to the astounding fact that certain topological invariants provide obstructions for geometric structures. Consequently , topological invariants of this type serve as interface...

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F -polynomials ∑ j fj−1((x − 1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev t...

Shifted Simplicial Complexes Are Laplacian Integral

We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial...