Niveau: Supérieur, Doctorat, Bac+8
SOME ABELIAN INVARIANTS OF 3-MANIFOLDS LOUIS FUNAR Some invariants for closed orientable 3-manifolds are defined using a series of representations of the symplec- tic groups and the theory of Heegaard splittings. They are natural extensions of the U(1) Chern-Simons- Witten invariants. These representations come from the functional equation satisfied by the theta functions of level k. We analyze the values of these invariants for lens spaces. Keywords: Theta functions, Heegaard splitting, tensor representation, symplectic groups. AMS Classification: 57 A 10, 14 K 25, 32 G 15. 0 . INTRODUCTION The aim of this paper is to construct invariants of 3-manifolds using the endomorphisms of 1-homologies of surfaces determined by Heegaard splittings and representations of the symplectic group. This leads us to the study of actions of such endomorphisms on the space of theta functions on the Siegel space. The construction goes as follows. Any three manifold can be given an Heegaard decomposition, and hence can be written as the union of two handlebodies identified along a homeo- morphism of the surface boundary. After a choice of a basis of the 1-homology the homeomorphism induces an element of the symplectic group. The indeterminacy in the choice of this matrix can be analyzed to give invariants of the three manifold in question. We develop a particular invariant using actions on spaces of modular forms and analyze it in the case of lens spaces.
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