O-minimal Cohomology and Definably Compact Definable Groups

Mário J. Edmundo

Let N an ℵ_{1}-saturated o-minimal expansion of a real closed field. We develop the theory of covers for N-spaces and the cohomology theory for the category of closed and bounded N-definable sets with continuous N-definable maps and use this to solve the problem of existence of torsion points on Ndefinably compact N-definable abelian groups. Namely we prove the following result: Let G be an Ndefinably compact N-definably connected N-definable group. Then the o-minimal Euler characteristic of G is zero. Moreover, if G is abelian and π_{1}(G) = ℤ^{n} ⊕ ℤ_{m_{1}^{l_{1}}} ⊕ ⋯ ⊕ ℤ_{m_{r}^{lr}}, then n > 0 and for all k > 1, card({x ∈ G : kx = 0}) = k^{n} · (k, m_{1}) ⋯ (k, m_{r}).

Address: Mathematical Institute, University of Oxford, 24-29 St Giles, OX1 3LB Oxford, U.K

E-mail: edmundo@maths.ox.ac.uk