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Realization of Hölder complexes. (English) Zbl 0954.14039

From the paper: Let \(\Gamma\) be a connected graph without loops, \(V_\Gamma= \{a_1,a_2, \dots,a_k\}\) be the set of vertices and \(E_\Gamma= \{g_1, g_2, \dots,g_r\}\) be the set of edges of the graph.
Definition: A Hölder complex \((\Gamma,\beta)\) is a graph \(\Gamma\) with an associated function \(\beta:E_\Gamma\to [1,\infty[ \cap\mathbb{Q}\).
Here we prove that each Hölder complex can be realized as a 2-dimensional semialgebraic set. For this purpose we embed the graph in an \(n\)-dimensional torus. The torus is vanishing in a singular point such that the generators are vanishing with different rational rates.

MSC:

14P10 Semialgebraic sets and related spaces
05C62 Graph representations (geometric and intersection representations, etc.)
14P05 Real algebraic sets
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References:

[1] Birbrair, L.) .- Local bi-Lipschitz classification of 2-dimensional semialgebraic sets, Preprint I.M.P.A. (1996). · Zbl 1007.32006
[2] Birbrair, L.) and Goldshtein, V.) .- An Example of Noncoincidence of Lp-cohomology and Intersection Homology for Real Algebraic varieties. I.M.R.N.6 (1994), pp. 265-271. · Zbl 0826.14011
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