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A classification of special 2–fold coverings

Anne Bauval, Daciberg L Gonçalves, Claude Hayat and Maria Hermínia de Paula Leite Mello

Geometry & Topology Monographs 14 (2008) 27–47

DOI: 10.2140/gtm.2008.14.27

arXiv: 0904.1194


Starting with an SO(2)–principal fibration over a closed oriented surface Fg, g≥ 1, a 2–fold covering of the total space is said to be special when the monodromy sends the fiber SO(2)∼ S1 to the nontrivial element of Z2. Adapting D Johnson's method [Spin structures and quadratic forms on surfaces, J London Math Soc, 22 (1980) 365-373] we define an action of Sp(Z2,2g), the group of symplectic isomorphisms of (H1(Fg;Z2),.), on the set of special 2–fold coverings which has two orbits, one with 2g-1(2g+1) elements and one with 2g-1(2g-1) elements. These two orbits are obtained by considering Arf-invariants and some congruence of the derived matrices coming from Fox Calculus. Sp(Z2,2g) is described as the union of conjugacy classes of two subgroups, each of them fixing a special 2–fold covering. Generators of these two subgroups are made explicit.


coverings, spin structures, quadratic forms, Fox calculus

Mathematical Subject Classification

Primary: 57R15

Secondary: 53C27


Received: 7 June 2006
Revised: 10 2007
Accepted: 16 January 2007
Published: 29 April 2008

Anne Bauval
Département de Mathématiques
Laboratoire Emile Picard, UMR 5580
Université Toulouse III
118 Route de Narbonne
31400 Toulouse
Daciberg L Gonçalves
Departamento de Matemática - IME-USP
Caixa Postal 66281
Agência Cidade de São Paulo
05311-970 - São Paulo - SP
Claude Hayat
Département de Mathématiques
Laboratoire Emile Picard, UMR 5580,
Université Toulouse III
118 Route de Narbonne
31400 Toulouse
Maria Hermínia de Paula Leite Mello
Departamento de Análise Matemática
Universidade Estadual do Rio de Janeiro
Rio de Janeiro