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Generating Simplicial Complexes

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 Added by Daniel Pons
 Publication date 2010
  fields
and research's language is English




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In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented.



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We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen--Flores theorem and provide a topological extension of the ErdH os--Ko--Rado theorem. By analogy with Farys theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.
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We estimate the upper bound for the $ell^{infty}$-norm of the volume form on $mathbb{H}^2timesmathbb{H}^2timesmathbb{H}^2$ seen as a class in $H_{c}^{6}(mathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R};mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by $mathbb{H}^{2}timesmathbb{H}^{2}timesmathbb{H}^{2}$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $big(mathbb{H}^2big)^n$.
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