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Register a new userRationality of a generating function of Betti numbers of the unordered configuration spaces of a punctured torus
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Given a torus $E = S^{1} times S^{1}$, let $E^{times}$ be the open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th singular Betti number $h^{i}(mathrm{Conf}^{n}(E^{times}))$ of the unordered configuration space of $n$ points on $E^{times}$ can be computed as a coefficient of an explicit rational function in two variables. Our proof uses Delignes mixed Hodge structure on the singular cohomology $H^{i}(mathrm{Conf}^{n}(E^{times}))$ with complex coefficients, by considering $E$ as an elliptic curve over complex numbers. Namely, we show that the mixed Hodge structure of $H^{i}(mathrm{Conf}^{n}(E^{times}))$ is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement will imply our main result about the singular Betti numbers. We also compute all the mixed Hodge numbers $h^{p,q}(H^{i}(mathrm{Conf}^{n}(E^{times})))$ as coefficients of an explicit rational function in four variables.
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Given a smooth quasiprojective variety $Y$ over $mathbb C$ that is not projective, consider its unordered configuration spaces $mathrm{Conf}^n(Y)$ for $ngeq 0$. Remove a point $P$ of $Y$ and obtain a one-puncture $Y-P$ of $Y$. We give a decomposition formula that computes the singular cohomology groups of $mathrm{Conf}^n(Y-P)$ in terms of those of $mathrm{Conf}^m(Y); (0leq mleq n)$, and prove it for several families of examples of $Y$, including the case where $Y$ is obtained from a smooth projective variety by puncturing one or more points. This formula keeps track of the mixed Hodge structures of the cohomology groups as well. This result simultaneously implies a result of Kallel involving Betti numbers and a consequence of a combinatorial property of configuration spaces due to Vakil and Wood. We also obtain intermediate results involving ordered configuration spaces that potentially work for more examples of $Y$.
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: ErdH{o}s-Renyi random clique complexes, random Vietoris-Rips complexes, and random v{C}ech complexes. These results may be of practical interest in topological data analysis.
We correct the proofs of the main theorems in our paper Limit theorems for Betti numbers of random simplicial complexes.
Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincare polynomials depend on the system of weights of the parabolic bundle.
We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $mathbb{R}mathrm{P}^n$. Using a random spectral sequence, we prove an asymptotically exact estimate on the expected number of connected components in the complement of $s$ such hypersurfaces in $mathbb{R}mathrm{P}^n$. We also investigate the same problem in the case where the hypersurfaces are defined by random quadratic polynomials. In this case, we establish a connection between the Betti numbers of such arrangements with the expected behavior of a certain model of a randomly defined geometric graph. While our general result implies that the average zeroth Betti number of the union of random hypersurface arrangements is bounded from above by a function that grows linearly in the number of polynomials in the arrangement, using the connection with random graphs, we show an upper bound on the expected zeroth Betti number of random quadrics arrangements that is sublinear in the number of polynomials in the arrangement. This bound is a consequence of a general result on the expected number of connected components in our random graph model which could be of independent interest.
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