Decomposability of homotopy lens spaces and free cyclic group actions on homotopy spheres

Abstract:

Let $\rho$ be a linear ${Z_n}$ action on ${{\text {C}}^m}$ and let $\rho$ also denote the induced ${Z_n}$ action on ${S^{2p - 1}} \times {D^{2q}},{D^{2p}} \times {S^{2q - 1}}$ and ${S^{2p - 1}} \times {S^{2q - 1}}$ where $p = [m/2]$ and $q = m - p$. A free differentiable ${Z_n}$ action $({\Sigma ^{2m - 1}},\mu )$ on a homotopy sphere is $\rho$-decomposable if there is an equivariant diffeomorphism $\Phi$ of $({S^{2p - 1}} \times {S^{2q - 1}},\rho )$ such that $({\Sigma ^{2m - 1}},\mu )$ is equivalent to $(\Sigma (\Phi ),A(\Phi ))$ where $\Sigma (\Phi ) = {S^{2p - 1}} \times {D^{2q}}{ \cup _\Phi }{D^{2p}} \times {S^{2q - 1}}$ and $A(\Phi )$ is a uniquely determined action on $\Sigma (\Phi )$ such that $A(\Phi )|{S^{2p - 1}} \times {D^{2q}} = \rho$ and $A(\Phi )|{D^{2p}} \times {S^{2q - 1}} = \rho$. A homotopy lens space is $\rho$-decomposable if it is the orbit space of a $\rho$-decomposable free ${Z_n}$ action on a homotopy sphere. In this paper, we will study the decomposabilities of homotopy lens spaces. We will also prove that for each lens space ${L^{2m - 1}}$, there exist infinitely many inequivalent free ${Z_n}$ actions on ${S^{2m - 1}}$ such that the orbit spaces are simple homotopy equivalent to ${L^{2m - 1}}$.