One fixed point actions on low-dimensional spheres (original) (raw)

Smooth compact Lie group actions on disks

Mathematische Zeitschrift, 1976

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).

Characteristic fixed-point sets of semifree actions on spheres

1999

A group action is semifree if it is free away from its fixed-point set. P. A. Smith showed that when a finite group of order q acts semifreely on a sphere, the fixed set is a mod q homology sphere. Conversely, given a mod q homology sphere as a subset of a sphere, one may try to construct a group action on the sphere fixing the subset. The converse question was first systematically studied by Jones and then by many others. In this note, we give new numerical congruences satisfied by the homology of the fixed sets and give a definitive solution to the problem for characteristic fixed-point sets.

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.

Fixed point free actions of spheres and equivariant maps

Topology and its Applications

This paper generalizes the concept of index and co-index and some related results for free actions of G = S 0 on a paracompact Hausdorff space which were introduced by Conner and Floyd[1]. We define the index and co-index of a finitistic free G-space X, where G = S d , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)sphere S 2n+1 (resp. (4n+3)-sphere S 4n+3) for the action of componentwise multiplication of G = S 1 (resp. S 3) is n. We also determine the orbit spaces of free actions of G = S 3 on a finitistic space X with the mod 2 cohomology and the rational cohomology product of spheres S n × S m , 1 ≤ n ≤ m. The orbit spaces of circle actions on the mod 2 cohomology X is also discussed. Using these calculation, we obtain an upper bound of the index of X and the Borsuk-Ulam type results.

$Z_2^2$-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.

Z2k-actions with a special fixed point set

Fundamenta Mathematicae, 2005

Let F n be a connected, smooth and closed n-dimensional manifold satisfying the following property: if N m is any smooth and closed m-dimensional manifold with m > n and T : N m → N m is a smooth involution whose fixed point set is F n , then m = 2n. We describe the equivariant cobordism classification of smooth actions (M m ; Φ) of the group G = Z k 2 on closed smooth m-dimensional manifolds M m for which the fixed point set of the action is a submanifold F n with the above property. This generalizes a result of F. L. Capobianco, who obtained this classification for F n = RP 2r (P. E. Conner and E. E. Floyd had previously shown that RP 2r has the property in question). In addition, we establish some properties concerning these F n and give some new examples of these special manifolds.