Hello, I want to show that the cohomological dimension (say over Z or R) of some group K is 1. K occurs in an exact sequence 1 -> K -> pi_1(X) -> pi_1(C) -> 1, where pi_1(X) has cohomological dimension 3 (in the same coefficients) and C is a curve of genus greater than 2.

So I want a kind of additivity but this is not true in general. If I look at the associated fibration BK -> Bpi_1(X) -> C and use Leray-Serre spectral sequence, I have some information on the cohomology of BK and in fact can solve the problem if I assume that the action of the fundamental group of B on the cohomology of the fiber is trivial. But I'm not familiar with cohomology with local coefficients and don't manage to show the general case.

Someone can help me ? (or solve this problem more directly ?) (or this is false in general ?)

mister_jones