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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Kai Wang
Journal: Trans. Amer. Math. Soc. 220 (1976), 361-371
MSC: Primary 57E25
DOI: https://doi.org/10.1090/S0002-9947-1976-0431237-4
MathSciNet review: 0431237
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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 )\vert{S^{2p - 1}} \times {D^{2q}} = \rho $ and $ A(\Phi )\vert{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}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0431237-4
Keywords: Wall groups, homotopy smoothings, normal invariants, decomposable homotopy lens spaces
Article copyright: © Copyright 1976 American Mathematical Society