The involution T: Cm+1 , Cm+' given by T(zo, z) = (zO, z) is invariant on Vq(n) and S2m_1, hence also invariant on Kq2m-1 preserving the obvious weakly complex structure. Moreover, T is fixed point free on Kq2m-1 because (z0, 0) cannot belong to both Vq(m) and S2m+1. The involutions (T, Kqm-1) will be referred to as the Brieskorn examples (cf. [3]). In [12] Milnor showed that (S2m+', K,2m-1) is a fibered pair over S' in the sense of Giffen [10], and each K247+1 is a homotopy sphere whose fibre in S4k+3 is a framed (4k + 2)-manifold with Arf invariant 0 if d. 0,3 mod4; 1 if d=_ 1, 2 mod4. In [11] we showed that if k > 1, (T, K24dk+') contains invariant codimension 1 and 2 spheres if and only if d 0_ , 3 mod 4; further, the value of the BrowderLivesay-Arf invariant [5] for (T, K2d+1) is zero if and only if d _ 0 3 mod 4. This provides a crude distinction of the Brieskorn examples in the piecewise linear (PL) category. In this paper we show that the weakly complex classification of these examples is considerably finer. Up to a choice of orientations, this is essentially a smooth distinction of the Brieskorn examples (cf. [1], [4]). These geometric results are studied via the bordism &(, (Z2) of fixed point free involutions preserving a weakly complex structure and the weakly complex cobordism Q*(P) of projective spaces. We first set up an integralization and extension of Conner's results on weakly complex bordism of involutions in [6; ?? 10-16, especially ? 16]. There results a surjective morphism of rings with unit