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On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus. (English) Zbl 1345.30058
Summary: In the moduli space $${\mathcal M}_{g}$$ of Riemann surfaces of genus $$g \geq 2$$ there is important, so-called, real locus $${\mathcal R}_{g}$$, consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. $${\mathcal R}_g$$ itself is covered by the strata $${\mathcal R}_g^{k}$$, each being formed by the points representing surfaces having a symmetry of given topological type $$k$$. These strata are known to be real analytic varieties of dimension $$3(g-1)$$. Also, their topological structure is pretty well known. Natanzon-Seppälä have realized that they are connected orbifolds homeomorphic to the factors of $$\mathbb{R}^{3g-3}$$ with respect to actions of some discrete groups and Goulden-Jackson-Harer and Harer-Zagier have found their Euler characteristics, expressing them through the Riemann zeta function. However, topological properties of the whole real locus $${\mathcal R}_{g}$$ were less studied. The most known fact is its connectivity, proved independently by Buser-Seppälä-Silhol, Frediani and Costa-Izquierdo. This paper can be seen as a further contribution to the study of topology of $${\mathcal R}_g$$, which was possible through the notion of the nerve $$\mathcal{N}_g$$, associated to $${\mathcal R}_{g}$$ and called the real nerve. We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of odd $$g$$. Precise values of these dimensions for even $$g$$ have been found by the authors in an earlier paper.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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