Smooth compact Lie group actions on disks (original) (raw)

1976, Mathematische Zeitschrift

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Abstract

In two earlier papers ([9] and [10]), the author studied smooth actions of finite groups on disks; in particular describing the fixed point sets (up to homotopy type) which can occur for such actions of any given finite group. The main result in was that for any finite group G not of prime power order, there exists an integer n o such that for any finite CW complex F, G has a smooth action on a disk with fixed point set having the homotopy type of F if and only if ;g(F) ~ 1 (mod no).

Replacement of fixed sets for compact group actions: The 2\rho theorem

2009

If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally linear actions on topological or PL manifolds by compact Lie groups, the replacement is always possible if the normal bundle of the fixed set is twice of a complex bundle over a 1-skeleton of the fixed set. Moreover, we also study some specific examples, where the answer to the replacement problem ranges from always possible to the rigidity.

Actions of some simple compact Lie groups on themselves

Proceedings - Mathematical Sciences, 2019

Let G be a compact connected simple Lie group acting non-transitively, non-trivially on itself. Hsiang (Cohomology theory of topological transformation groups, (1975) (New York: Springer)) conjectured that the principal isotropy subgroup type must be the maximal torus and the action must be cohomologically similar to the adjoint action and the orbit space must be a simplex. But Bredon (Bull AMS 83(4) (1977) 711-718) found a simple counterexample, where the principal isotropy subgroup is not a maximal torus and which has no fixed point. In this work, we prove that if SO(n), (n ≥ 34) or SU (3) acts smoothly (and nontrivially) on itself with non-empty fixed point set, then the principal isotropy subgroups are maximal tori.

One fixed point actions on low-dimensional spheres

Inventiones Mathematicae, 1990

When one studies the symmetry groups of spheres, disks and Euclidean spaces, it is often very fruitful to begin by comparing the properties of linear symmetry groups with those of more general examples. In particular, if the fixed point set is finite, it is natural to ask if the number of fixed points coincides with the number for some linear action (namely, 0 or 2 for actions on spheres and 1 for actions on disks and Euclidean spaces). During the nineteen forties P.A. Smith, D. Montgomery and H. Samelson raised questions about the existence of compact Lie group actions on spheres with one fixed point and on disks and Euclidean spaces with no fixed points (see [Eil], Problem 39, and [MSa], Section 7). Recently such actions have attracted additional interest in connection with regularity conjectures for algebraic group actions on affine n-space (see [PR]). Of course, the questions for various spaces are related. In particular, continuous fixed point free actions on Euclidean spaces correspond bijectively to one fixed point actions on spheres via one point compactification.

Group actions having one fixed point

Mathematische Zeitschrift, 1986

in 1983. Question. Which compact Lie groups G can act smoothly (and effectively) on a closed (oriented) manifold M" of positive dimension so that the fixed point set M G consists of precisely one point ? Comments. 1. There is no loss in presuming M is connected. The component of M containing the fixed point will be invariant under the action of G. 2. If M is oriented and connected, then the diagonal action of G on M x M, g(m,m')=(gm, gm'), also fixes precisely one point, and will automatically preserve orientation. Thus, one need have no concern about preservation of orientation.

Deleting-Inserting Theorem for smooth actions of finite nonsolvable groups on spheres

Commentarii Mathematici Helvetici, 1995

The paper presents a method which allows to construct smooth finite nonsolvable group actions on spheres with prescribed fixed point data. The idea is to consider an action on a disk with the required fixed point data, and then to apply equivariant surgery to the equivariant double of the disk to remove the second copy of the fixed point data. In this paper, the method is applied to construct smooth group actions on spheres with exactly one fixed point, and more general actions with fixed point set diffeomorphic to any given closed stably parallelizable smooth manifold. The method is expected to be useful for constructions of smooth group actions on spheres with more complicated fixed point data.

Towards the solution of some fundamental questions concerning group actions on the circle and codimension-one foliations

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due toÉ. Ghys, G. Hector and D. Sullivan.

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References (12)

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Boothby, W., Wang, H.-C.: On the finite subgroups of connected Lie groups. Commentarii math. Helvet. 39, 281-294 (1964)
Conner, P.: Retraction properties of the orbit space of a topological transformation group. Duke math. J. 27, 341-357 (1960)
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tom Dieck, T.: The Burnside ring of a compact Lie group I. Math. Ann. 215, 235-250 (1975)
Illman, S.: Equivariant algebraic topology. Thesis, Princeton University (1972)
Montgomery, D., Zippin, L.: Topological transformation groups. New York-London: Inter- science 1955
Nakayama, T.: Modules of trivial cohomology over a finite group II. Nagoya math. J. 12, 171-176 (1957)
Oliver, R.: Fixed point sets of group actions on finite acyclic complexes. Commentarii math. Helvet. 155-177 (1975)
Oliver, R.: Projective obstructions to group actions on disks. Preprint No. 17. Matematisk Institut: Aarhus 1974/75
Rotman, J.: The theory of groups. Allyn and Bacon, 1965
Swan, R.: Induced representations and projective modules. Ann. Math., II. Ser. 71, 552-578 (1960)

Replacement of fixed sets for compact group actions: The 2ρ theorem

2010

Fixed Point Sets and the Fundamental Group I: Semi-free Actions on G-CW-Complexes

2020

Smith theory says that the fixed point of a semi-free action of a group G on a contractible space is Z p-acyclic for any prime factor p of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain K-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the K-theoretical obstruction.

The localization theorem for finite-dimensional compact group actions

TURKISH JOURNAL OF MATHEMATICS

The localization theorem is known for compact G-spaces, where G is a compact Lie group. In this study, we show that the localization theorem remains true for finite-dimensional compact group actions, and Borel's fixed point theorem holds not only for torus actions but for arbitrary finite-dimensional compact connected abelian group actions.

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

Transactions of the American Mathematical Society, 1983

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.

$Z_2^2$-actions with nnn-dimensional fixed point set

Proceedings of the American Mathematical Society, 2007

We describe the equivariant cobordism classification of smooth actions (M m , Φ) of the group G = Z 2 2 , considered as the group generated by two commuting involutions, on closed smooth m-dimensional manifolds M m , for which the fixed point set of the action is a connected manifold of dimension n and m = 4n − 1 or 4n − 2. For m ≥ 4n, the classification is known. 1991 Mathematics Subject Classification. (2.000 Revision) Primary 57R85; Secondary 57R75. Key words and phrases. Z 2 2-action, fixed data, equivariant cobordism class, characteristic number, projective space bundle, Stiefel-Whitney class. The author was partially supported by CNPq and FAPESP.

The topological spherical space form problem—II existence of free actions

Topology, 1976

RECENT advances in calculation of projective class groups and of surgery obstruction groups lead us to hope that it will shortly be possible to give a fairly complete account of the classification of free actions of finite groups on spheres. In the present paper, we determine which groups can so act, thus solving a problem of several years' standing. Further, we show that these actions can be taken to be smooth actions on smooth homotopy spheres. Previously known results can be summarised as follows, where we say the finite group 7~ satisfies the "pq-condition" (p, q primes not necessarily distinct) if all subgroups if v of order pq are cyclic. 0.1. (Cartan and Eilenberg[3]). If rr acts freely on S"-', it has periodic cohomology with minimum period dividing n. Moreover, P has periodic cohomology if and only if it satisfies all p2-conditions. And the p* condition is equivalent to the Sylow p-subgroup zrr, of r being cyclic or perhaps (if p = 2) generalised quaternionic. 0.2. (Wolf [19]). If 7~ acts freely and orthogonally on a sphere, it satisfies all pq-conditions. Conversely, if r is soluble and satisfies all pq-conditions, free orthogonal actions exist. However, for rr non-soluble, the only non-cyclic composition factor allowed is the simple group of order 60. 0.3. (Milnor [9], see also Lee [8]). If 7~ acts freely on any sphere, it satisfies all 2p-conditions. 0.4. (Petrie [I 11). Any extension of a cyclic group of odd order m by a cyclic group of odd prime order q prime to m can act freely on S*"-'. Petrie's result shows that pq-conditions are not all necessary for free topological actions. it is therefore not so surprising that THEOREM 0.5. A finite group 7 can act freely on a sphere if and only if it satisfies all 2p-and p '-conditions. We shall elaborate the group theory in the next section: the most interesting groups 7r included are perhaps the groups S&(p) (p prime). As to smooth actions, it will follow from a general result below that THEOREM 0.6. For each free action of T on S"-' constructed in the proof of (0.5). S"-' has a differential structure (z such that rr acts freely and smoothly on S,"-'. Clearly, in many cases one can deduce existence of free smooth actions on S"-', but in this paper we will confine ourselves to general arguments. In principle, the proof of these theorems follows the pattern laid down in [ 151 and elaborated for this problem in a previous paper[l4]. We construct first, a finite simplicial complex X; second, a normal invariant, and hence a normal cobordism class of normal maps M+X; and thirdly we show that the corresponding surgery obstruction vanishes. This yields a manifold homotopy equivalent to X whose universal cover is homotopy equivalent, hence homeomorphic (but not necessarily diffeomorphic, when smooth) to a sphere. In practice, the key idea of the proof is a careful choice of X, and of the normal invariant, so as to allow a simple proof of vanishing of the suergery obstruction. The paper is set out in four sections. In the first, we summarise the group theory, and introduce notations for the groups involved. In the second, we choose the homotopy type of X: this involves circumventing the finiteness obstruction of Swan[ 131, and prepares the way for the surgery. In the third, we discuss normal invariants. General existence of normal invariants follows from the powerful techniques of modern homotopy theory. Topological normal invariants can then be studied using Sullivan's[lO] analysis of the homotopy type of G/Top.

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Invariant topological complexity

Bulletin of the London Mathematical Society, 2014

We present a new approach to an equivariant version of Farber's topological complexity called invariant topological complexity. It seems that the presented approach is more adequate for the analysis of impact of a symmetry on a motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the invariant topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the topological complexity of the orbit space. We define the Whitehead version of it.

Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

Algebraic & Geometric Topology, 2002

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a G ≥ 2 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a G ≥ 2. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a G = 0 or 1.

Fixed point sets and the fundamental group II: Euler characteristics

Proceedings of the Royal Society of Edinburgh: Section A Mathematics

For a finite group GGG of not prime power order, Oliver showed that the obstruction for a finite CW-complex FFF to be the fixed point set of a contractible finite GGG -CW-complex is determined by the Euler characteristic chi(F)\chi (F)chi(F) . (He also has similar results for compact Lie group actions.) We show that the analogous problem for FFF to be the fixed point set of a finite GGG -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on K_0K_0K_0 [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

Smooth compact Lie group actions on disks (original) (raw)

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