Construction of Aspherical Manifolds from Special $G$-Manifolds
Hiroshi Nakamura · Tokyo Journal of Mathematics · 1982
Communicated by T. Mitsui)$M$ (roughly speaking, isotropy groups of $G$ at $x\in X$ ).Note that the following fact is known: If $G$ is abelian, then $S[U_{A}]\cong\prod[G]\cong[M;BG]$ (see [6, Corollary 1]).That is, the isomorphic class [X] depends only on the isomorphic class of the G-principal bundle $P$ , and the class [X] corresponds to a homotopy class of maps of $M$ into the classifying space $BG$ .But actually the homotopy groups of $X$ can not be computed directly even if the homotopy groups of $M$ are com- putable.In general also we do not know whether this $X$ is an aspherical (i.e., its universal covering is contractible) manifold or not.In this paper we give a condition that the special G-manifold is aspherical.In this case it is known from the result of Conner and Raymond [1, Theorem 5.6] that $G$ is a toral group and all isotropy groups are finite.And under this condition it follows from Lemma 1 that the orbit structure $U_{4}$ over $M$ is a family of $U_{\alpha}$ which is isomor $\cdot$ phic to $Z_{2}$ .And our main result is the following THEOREM 1.Let $T^{k}$ be a k-dimensional toral group $(k>0),$ $M^{n}$ an m-dimensional compact connected diferentiable manifold with boundary $\partial M=\bigcup_{\alpha eA}B_{\alpha}$ , where $B_{\alpha}$ is a connected component $(m>0)$ .Let $(Z_{2})_{A}=$