Euler Class and Free Generation

This paper consists of two parts. In the first auxilliary part, we deal with sets with cyclic order. For any such set O we introduce following [BG] a cocycle l : G × G → Z valued in {0,±1} ⊂ Z on the group G of automorphisms of O. The cohomology class of l in H(G,Z) will be called the Euler class. If K is an ordered field, then the projective line P(K) has a cyclic order and PSL2(K) acts orderpreserving on P(K), so that we get both the cocycle l and the Euler class in H(PSL2(K),Z). If K = R, then our Euler class coincides with the usual Euler class on PSL2(R). In view of our extension of the Euler class to all ordered fields, the following two problems arise.