Free Actions Of Finite Groups Research Articles

Extending free actions of finite groups on surfaces

The present work introduces new perspectives in order to extend orientation-preserving finite group actions from oriented surfaces to 3-manifolds. We consider the Schur multiplier associated to a finite group G in terms of two-dimensional principal G-bordisms, called G-cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is always affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show some particular cases for non-necessarily free actions of abelian groups and dihedral groups on surfaces.

Homotopy classification of actions of finite groups on aspherical spaces

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.

Classification of free actions on complete intersections of four quadrics

In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in $\PP^7$, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.

Free actions of finite groups on $S^n \times S^n$

Let p be an odd prime. We construct a non-abelian extension Γ of S by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.

Free actions of finite groups on the 3-dimensional nilmanifold

We study free actions of finite groups on the standard 3-dimensional nilmanifold. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying normal nilpotent subgroups of all almost Bieberbach groups of finite index, up to affine conjugacy. We conclude that if a finite group acts freely on the standard 3-dimensional nilmanifold with the first homology Z 2 , then it is cyclic.

Classification of free actions of finite groups on the 3-torus

We classify free actions of finite groups on the 3-torus, up to topological conjugacy. By works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal free Abelian subgroups of affine Bieberbach groups with finite quotients, up to affine conjugacy. A complete classification of all such groups is obtained by utilizing Lee–Shin–Yokura's method [Topology Appl. 53 (1993) 153–175]. Thereby this gives a complete classification of free actions of finite groups on the 3-torus, up to topological conjugacy.

Free actions of finite groups on rational homology 3-spheres

We show that any finite group can act freely on a rational homology 3-sphere.

Complex surfaces which are fibre bundles

We characterize complex surfaces admitting holomorphic submersions to complex curves and quotients of such surfaces by free actions of finite groups in terms of their Euler characteristics and fundamental groups.

Free actions of finite groups on products of symmetric powers of even spheres

This paper answers a question on the existence of free actions on products of symmetric powers of even-spheres. The main objective is to show that a finite group G acts freely on a finite product of symmetric powers of even-dimensional spheres iff it can act freely on a suitable product of evendimensional spheres themselves.

The $C^1$ closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups

The $C^1$ closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups

Extending finite group actions from surfaces to handlebodies

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

Local surgery obstructions and space forms

The space form problem, concerning the existence of free actions of finite groups on spheres, has now been substantially solved except in dimension three I-3, 4, 8, 11, 13, 14]. This paper gives an effective method for the uniqueness question: which homotopy types are realized by such actions. Our results are fairly complete for 2-hyperelementary periodic groups and give new necessary conditions for actions even on S 3. In particular Corollary C below eliminates certain homotopically non-linear actions which presented an obstacle to knowing precise dimensional bounds (see [4; Conjecture D]). The methods also apply to the euclidean space form problem of semi-free actions on I( "+k fixing I( k [2] where the linear model is a free representation direct sum with a trivial representation. A free action on S'- 1 yields a semi-free topological action on l(" by "coning". The method involves the calculation of a certain local surgery obstructions (w167 2, 3). The basic cases are the 2-hyperelementary type I groups: extensions of the form l ~ Z/m~ ~--, Z/2k--> l where m is odd. We write o- for a Sylow 2-subgroup of ~z (and identify it with Z/2k). Let t be the twisting defining the extension, and let t(cr) = Im(t: Z/2 k --* Aut(Z/m)) -~ 7/./2 z.

A Note on Simple Duality

Certain spaces satisfying Poincaré duality are shown to have the homotopy type of simple Poincaré complexes. The motivation for and applications of this result are the study of free actions of finite groups on manifolds.

A note on simple duality

Certain spaces satisfying Poincaré duality are shown to have the homotopy type of simple Poincaré complexes. The motivation for and applications of this result are the study of free actions of finite groups on manifolds.

The Surgery Semicharacteristic

Given a degree-one normal map ( / , / ) : (M,vM) -> (X, £), C. T. C. Wall defined the associated surgery obstruction a(f, f) e L^JLn^X). This obstruction vanishes if and only if (/, / ) is normally bordant to a simple homotopy equivalence. Using Wall's approach, one must perform preliminary surgery to make / highly connected before the obstruction can be calculated. It is natural to ask for invariants which can be calculated without preliminary surgery. In the even-dimensional case the multisignature gives such an invariant. The purpose of this paper is to present an odd-dimensional normal bordism invariant defined without preliminary surgery. This invariant is analogous to the semicharacteristic bordism invariant introduced by Lee [13]. A justification for the normal bordism invariant is that it gives strong restrictions on the homology of a manifold with a free action of a finite group. In particular, it gives a good explanation of Lee's results on free actions of finite groups on spheres. Let I be a (2n + l)-dimensional Poincare complex with fundamental group n. Suppose A is a semisimple ring with involution a \-> a, for a e A. Then L2n + 1(A) can be identified as a quotient of a subgroup of ^ 0 (^ ) ( §2). Given a homomorphism (Zn,co) -> (A, —), we can regard A as a local coefficient system (that is, a Znmodule). The surgery semicharacteristic is

Erratum to: Free actions of finite groups on varieties. II

Erratum to: Free actions of finite groups on varieties. II

Free actions of finite groups on varieties. II

Free actions of finite groups on varieties. II

Free actions of finite groups on finite CW complexes

Free actions of finite groups on finite CW complexes

Factorizations of Free Actions of Finite Groups on Compact Q-Manifolds

Factorizations of Free Actions of Finite Groups on Compact Q-Manifolds

Factorizations of free actions of finite groups on compact $Q$-manifolds

Factorizations of free actions of finite groups on compact $Q$-manifolds