In 1980 and 1983, it was proved that P

-groups are surface groups ([2], [3]). Since then, topologists have been positively studying about P

-groups (or

-groups). For example, let a topological space X have a right .pi.-action, where .pi. is a multiplicative group. If each x.memX has an open neighborhood U such that for each u.mem..pi., u.neq.1, U.cap.

=.phi., this right .pi.-action is said to be proper. In this case, if X/.pi. is compact then (1) .pi.

(X/.pi).iden..pi.(X:connected, .pi.

: fundamental group) ([4]), (2) if X is a differentiable orientable manifold with demension n and .rho.X (the boundary of X)=.phi. then

(X;Z).iden.

(X;Z), ([6]), where Z is the set of all integers.s.