In the moduli space of Riemann surfaces of genus there is important, so-called, real locus , consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. {itself} is covered by the strata , each being formed by the points representing surfaces having a symmetry of given topological type . These strata are known to be real analytic varieties of dimension . Also, their topological structure is pretty well known. Natanzon–Seppälä have realized that they are connected orbifolds homeomorphic to the factors of 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 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 , {which was possible through the notion of the nerve , associated to 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 . Precise values of these dimensions for even have been found by the authors in an earlier paper.
Mots-clés : Riemann surface, moduli space of Riemann surfaces, real locus, symmetry of Riemann surface, oval of a symmetry of a Riemann surface, separability of a symmetry of Riemann surface, Fuchsian groups, NEC groups
@article{RSMUP_2016__135__91_0, author = {Gromadzki, Grzegorz and Koz{\l}owska-Walania, Ewa}, title = {On dimensions of the real nerve of the moduli space of {Riemann} surfaces of odd genus}, journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova}, pages = {91--109}, publisher = {European Mathematical Society Publishing House}, address = {Zuerich, Switzerland}, volume = {135}, year = {2016}, doi = {10.4171/RSMUP/135-5}, url = {http://www.numdam.org/articles/10.4171/RSMUP/135-5/} }
TY - JOUR AU - Gromadzki, Grzegorz AU - Kozłowska-Walania, Ewa TI - On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2016 SP - 91 EP - 109 VL - 135 PB - European Mathematical Society Publishing House PP - Zuerich, Switzerland UR - http://www.numdam.org/articles/10.4171/RSMUP/135-5/ DO - 10.4171/RSMUP/135-5 ID - RSMUP_2016__135__91_0 ER -
%0 Journal Article %A Gromadzki, Grzegorz %A Kozłowska-Walania, Ewa %T On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus %J Rendiconti del Seminario Matematico della Università di Padova %D 2016 %P 91-109 %V 135 %I European Mathematical Society Publishing House %C Zuerich, Switzerland %U http://www.numdam.org/articles/10.4171/RSMUP/135-5/ %R 10.4171/RSMUP/135-5 %F RSMUP_2016__135__91_0
Gromadzki, Grzegorz; Kozłowska-Walania, Ewa. On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus. Rendiconti del Seminario Matematico della Università di Padova, Tome 135 (2016), pp. 91-109. doi : 10.4171/RSMUP/135-5. http://www.numdam.org/articles/10.4171/RSMUP/135-5/
Cité par Sources :