Semifree Actions Research Articles

Semi-free group actions

Semi-free group actions

Semifree Actions of Finite Groups on Homotopy Spheres

Semifree Actions of Finite Groups on Homotopy Spheres

Semifree actions on finite groups on homotopy spheres

We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial.

Differentiable group actions on homotopy spheres

In several previous papers numerous results conserning smooth group actions on homotopy spheres not bounding parallelizable manifolds have been obtained [5, 13, 21, 23-26, 31, 32, 35, 39, 42-50]. This paper is a first step towards placing such results in a more unified setting; we shall concentrate our attention on semifiee actions to avoid book-keeping problems with various orbit types, postponing these considerations to later papers in the series. When combined with appropriate homotopy-theoretic machinery, this setting yields new proofs and substantial strengthenings of many known results. The basic underlying problem of this paper is the following: Given a homotopy (n +2k)-sphere 22, does it admit a semifree G-action with an n-dimensional fixed point set'? Our techniques show that the answer depends on the decomposition of the Pontrjagin-Thom invariants of 22 (which form a coset in the stable homotopy of spheres) in terms of lower-dimensional stable homotopy classes. In a subsequent paper we shall illustrate this by studying certain low-dimensional homotopy spheres in detail, obtained virtually complete information. On the other hand, the new non-existence theorems obtained in this paper also fall into this pattern indirectly; for if one knows that some specific composition operation (ordinary or higher order) on a stable homotopy class is nontriviat, such information often implies the class cannot be decomposed in the "right" way for an action to exist. Presumably some appropriate global information about the stable homotopy of spheres (perhaps resembling the Kahn-Priddy Theorem) is needed to fully answer questions such as the existence of smooth Zp actions on every homotopy sphere. The general viewpoint of this paper is derived from the work of Browder, Petrie, Rothenberg, and Sondow that classified certain semifree actions on homotopy spheres ([8, 10, 11, 39, 40]; also see [48]). A general motivating problem was the determination of the underlying differential structure on the ambient G-manifold from the classification information presented in the above papers; the insight needed to study this problem evolved from consideration of the examples in [45-47] where the differential structure was determined by entirely different methods. Our basic result (Theorem 3.4) establishes a strong relationship between the differentiable structure on 2" and an invariant that vaguely measures the knottedness of the fixed point set (the knot invariant of Section 2, which agrees with the internal normal invariant of [49]). This relationship exists because the

Applications of intersection theory

A Borsuk-Ulam type theorem for semifree actions on S 2 n {S^{2n}} is obtained via a formula of Lefschetz.

Semifree Actions on Homotopy Spheres

Semifree Actions on Homotopy Spheres

Semifree actions on homotopy spheres

In this paper, we study the semifree Z m {Z_m} actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree Z 4 s {Z_{4s}} actions on even dimensional homotopy sphere pairs.

Diffeomorphisms of manifolds and semifree actions on homotopy spheres

Diffeomorphisms of manifolds and semifree actions on homotopy spheres

A Note on Semifree Actions of S 1 on Homotopy Spheres

A Note on Semifree Actions of S 1 on Homotopy Spheres

Fixed-point sets of transformation groups on infinite-product spaces.

then a is said to be a semifree action of G on X with fixed-point set Y. If Y is empty, then a is said to be free. The problem addressed in this paper is to find sufficient conditions that each closed subset of X be the fixed-point set of a semifree action of G. The main result proven here is Theorem 3.2: If X is a separable, infinite-dimensional Fr&chet space and G is a topological group for which there is an effective action on X as a transformation group, then each closed subset of X is the fixedpoint set of a semifree action of G on X. This generalizes the theorem of V. L. Klee, Jr. [8] that each compactum of a separable, infinitedimensional Hilbert space is the fixed-point sef of periodic homeomorphisms of the space onto itself of all periods greater than one, and is complementary to A. Beck's theorem [4] that if X is a metric space on which there is an action of the real line without degenerate orbits, then each closed set of X is the union of the degenerate orbits of some action of the real line on X (in the sense that whereas Beck's theorem removes restrictions on the space X, the theorem of this paper removes restrictions on the group G).

A note on semifree actions of 𝑆¹ on homotopy spheres

A note on semifree actions of 𝑆¹ on homotopy spheres

Some Remarks on Circle Action on Manifolds

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition $\lambda=(\lambda_{1},...,\lambda_{u})$ of weight $m$ such that the Chern number $(c_{\lambda_{1}}... c_{\lambda_{u}})^{n}[M]$ (resp. Pontrjagin number $(p_{\lambda_{1}}... p_{\lambda_{u}})^{n}[N]$) is nonzero, then \emph{any} circle action on $M^{2mn}$ (resp. $N^{4mn}$) has at least $n+1$ fixed points. When an even-dimensional smooth manifold $N^{2n}$ admits a semi-free action with isolated fixed points, we show that $N^{2n}$ bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with \emph{nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.