Free Actions and Complex Cobordism

Abstract

Connor and Floyd have observed that a free action of a finite group $G$ on a compact manifold $M$ preserving a stable almost complex structure produces a stably almost complex quotient manifold $M/G$. Hence, the bordism group of such actions, $U_ \ast ^{G,{\text {free}}}$, is just ${U_ \ast }(BG)$. If $G$ is not finite or abelian, but an arbitrary compact Lie group, the tangent bundle along the fibres gives trouble. Nevertheless, it is shown that if ${H^ \ast }(BG)$ is torsion free then $U_ \ast ^{G,{\text {free}}} \approx {U_ \ast }(BG)$.