Basis of homology adapted to the trigonal automorphism of a Riemann surface. (English) Zbl 1149.30032

Campos, Helena B.

RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 101, No. 2, 167-172 (2007).

A compact Riemann surface \(S\) of genus \(g\) is said to be cyclic trigonal if there exists a conformal automorphism \(\varphi\) of order \(3\), called a trigonal morphism, such that the quotient space \(S/\langle \varphi \rangle\) has genus \(0\). Such surfaces can be thought of as generalizations of hyperelliptic surfaces and they have been the focus of a number of studies, see for example [R. D. M. Accola, Trans. Am. Math. Soc. 283, 423-449 (1984, Zbl 0584.14016) and Kodai Math. J. 23, No.1, 81-87 (2000, Zbl 0966.14021), A. Costa, M. Izquierdo and D. Ying [Manuscr. Math. 118, No. 4, 443-453 (2005, Zbl 1137.30013)] and A. Costa, and M. Izquierdo [Math. Scand. 98, No. 1, 53-68 (2006, Zbl 1138.30023)]. In the paper under review, the author determines a basis \(\mathcal{B}\) for the first homology group of a cyclic trigonal surface \(S\) which is adapted to a trigonal morphism \(\varphi\), (meaning the representation of the action of \(\varphi\) on the first homology group is very simple), and then determines the matrix of the intersection form on this basis. As an application, the author considers the problem of topological classification of automorphism groups of Riemann surfaces. The approach to the construction of a basis is as follows. The author considers a number of specially constructed paths on the surface \(S/C\) around the branch points of the map \(\pi \colon S\rightarrow S/C\). This gives rise to specific elements of the fundamental group of \(S/C\) and correspondingly to specific elements of the fundamental group of \(S\). The author is then able to use these elements to construct a basis \(\mathcal{B}\) for the first homology group of \(S\), which by construction, provides a simple representation of the action of \(\varphi\) on the first homology group of \(S\). With this explicit construction, the author is then able to determine the intersection form on \(S\) with respect to this basis.

Reviewer: Aaron Wootton (Portland)

MSC:

30F30	Differentials on Riemann surfaces
14H30	Coverings of curves, fundamental group
14H37	Automorphisms of curves

Keywords:

Cyclic trigonal surface; first homology group

Citations:

Zbl 0584.14016; Zbl 0966.14021; Zbl 1137.30013; Zbl 1138.30023