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Homological connectivity of random $$k$$-dimensional complexes. (English) Zbl 1177.55011
The authors study the question of homological $$(k-1)$$-connectivity of random $$k$$-dimensional complexes for a general $$k$$. The one-dimensional case (that is, connectivity of random graphs) was considered in [P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci., Ser. A 5, 17–61 (1960; Zbl 0103.16301)], and the two-dimensinal case was considered in [N. Linial and R. Meshulam, Combinatorica 26, No. 4, 475–487 (2006; Zbl 1121.55013)].
To describe the result obtained in the paper under review, let $$\Delta_{n-1}$$ denote the $$(n-1)$$-dimensional simplex. The authors consider the space $$Y_k(n,p)$$ of complexes $$\Delta_{n-1}^{(k-1)}\subset Y\subset \Delta_{n-1}^{(k)}$$ equipped with some adapted probability measure. A $$(k-1)$$-simplex $$\sigma\in \Delta_{n-1}(k-1)$$ is said to be isolated in $$Y$$ if it is not contained in any of the $$(k-1)$$-simplices of $$Y$$. For such an isolated simplex $$\sigma$$, its indicator function is a non-trivial $$(k-1)$$-cocycle of $$Y$$, hence $$H^{k-1}(Y)\not=0$$. The main result of the paper says that the threshold probability for the vanishing of $$H^{k-1}(Y)$$ coincides with the threshold for the non-existence of isolated $$(k-1)$$-simplices in $$Y$$.

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 60B99 Probability theory on algebraic and topological structures 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
random complex; homological connectivity
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##### References:
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