Characteristic invariants of free differentiable actions of S1 and S3 on homotopy spheres

Ku, Hsu-Tung; Ku, Mei-Chin

doi:10.1007/bfb0070032

Proceedings of the Second Conference on Compact Transformation Groups

1972

DOI: 10.1007/bfb0070032

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Characteristic invariants of free differentiable actions of S1 and S3 on homotopy spheres

Hsu-Tung Ku

Mei-Chin Ku

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1974

1983

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Cited by 5 publications

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“…Invariant codimension 2 subspheres have been much studied, particularly in the free case; papers of Browder [3], Cappell and Shaneson [12,13], Homer [17,18], H. T. and M. C. Ku [20], Montgomery and Yang [30] and Stoltzfus [55] give a representative sampling of such results.…”

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confidence: 99%

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Schultz¹

1983

Trans. Amer. Math. Soc.

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Abstract. A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two subsphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic 10-sphere admits a smooth circle action.In our previous papers in this series [45,44], we have considered the theory of semifree actions on homotopy spheres as formulated by W. Browder and T. Pétrie [10] and M. Rothenberg and J. Sondow [34]. Specifically, in the first paper a method was presented for describing (at least formally) those exotic spheres admitting such semifree actions-a problem first posed explicitly by Browder in [3, Problem l,p. 7] -and the second paper extended the whole theory to handle certain actions that are not semifree. This paper will treat another problem posed in Browder's paper [3, Problem 3] regarding invariant subspheres of homotopy spheres with group actions.One motivation for considering this question is that linear actions on spheres generally admit a great assortment of invariant linear subspheres (e.g., if the group is abelian and the dimension is much larger than the group's order), and from this viewpoint the existence of invariant subspheres reflects the extent to which an arbitrary smooth action resembles some natural linear model. In particular, this idea is central to the work of Browder and Livesay on free involutions [8] (compare also [3]).The existence of such subspheres is directly related to the realizability of actions as equivariant smooth suspensions, providing basic necessary conditions for such a realization. We shall also consider a dual problem in this paper; namely, the description of those group actions that can be smoothly equivariantly suspended.

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confidence: 99%

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Schultz¹

1983

Trans. Amer. Math. Soc.

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“…Actions on homotopy ll-spheres. In [8] we studied the differentiable actions of S3 on homotopy ll-spheres and proved Theorem 3.1. Let EJ,1 denote the Milnor sphere which represents the generator of 011 [9].…”

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“…The proof of this theorem is based on the examples constructed by the Hsiang brothers [7] and the following fact [8]: Let S3 act freely and differentiably on a homotopy ll-sphere X11. Then either where ä denotes the generator of ZZ4(S11/S3, Z) and ^(S11) the Eells-Kuiper /¿-invariant [4].…”

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confidence: 99%

“…Then either where ä denotes the generator of ZZ4(S11/S3, Z) and ^(S11) the Eells-Kuiper /¿-invariant [4]. It is said in [8] that the homotopy sphere kH]}, k odd, admits no free differentiable ^-actions. Since we made some errors in computation, this does not follow from Lemma 2.1 of [8].…”

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confidence: 99%

“…It is said in [8] that the homotopy sphere kH]}, k odd, admits no free differentiable ^-actions. Since we made some errors in computation, this does not follow from Lemma 2.1 of [8]. We conjecture that this result remains true.…”

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confidence: 99%

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Characteristic spheres of free differentiable actions of 𝑆¹ and 𝑆³ on homotopy spheres

Ku¹,

Ku²

1971

Trans. Amer. Math. Soc.

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Free 𝑆¹ actions and the group of diffeomorphisms

Wang¹

1974

Trans. Amer. Math. Soc.

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Let S 1 {S^1} act linearly on S 2 p − 1 × D 2 q {S^{2p - 1}} \times {D^{2q}} and D 2 p × S 2 q − 1 {D^{2p}} \times {S^{2q - 1}} and let f : S 2 p − 1 × S 2 q − 1 → S 2 p − 1 × S 2 q − 1 f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}} be an equivariant diffeomorphism. Then there is a well-defined S 1 {S^1} action on S 2 p − 1 × D 2 q ∪ f D 2 p × S 2 q − 1 {S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}} . An S 1 {S^1} action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.

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Characteristic invariants of free differentiable actions of S1 and S3 on homotopy spheres

Cited by 5 publications

References 12 publications

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Characteristic spheres of free differentiable actions of 𝑆¹ and 𝑆³ on homotopy spheres

Free 𝑆¹ actions and the group of diffeomorphisms

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