We study finite abelian group actions on weakly Lefschetz cohomologically symplectic (WLS) manifolds, a collection of manifolds that includes all compact connected Kaehler manifolds. We prove that for any WLS manifold X there exists a number C such that, for any integer m≥ C, if (𝐙/m)^k acts freely on X, then ∑_j b_j(X;𝐐)≥ 2^k. We also prove a structure theorem for effective actions on WLS manifolds of (𝐙/p)^r, where p is a big enough prime, analogous to some results for tori of Lupton and Oprea, and we find bounds on the discrete degree of symmetry of WLS manifolds. Our technique, which may be of independent interest, is based on studying the cohomology of abelian covers of WLS manifolds X associated to certain maps π:X→ T^k. We prove that, in the presence of actions of arbitrarily big finite abelian groups, some of these abelian covers have finitely generated cohomology, and the spectral sequence associated to π degenerates at the second page over the rationals.

We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete degree of symmetry. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold X such that H-& lowast; ( X ; Z ) is finitely generated, we prove the existence of a constant C with the property that for any continuous action of a finite group G on X such that every g is an element of G fixes at least one point of X , there is a subgroup H <= G satisfying [ G : H ] <= C and a point x is an element of X which is fixed by all elements of H .

In this paper we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group $G$ on a manifold $X$, these results provide information on the restriction of the action to a subgroup of $G$ of index bounded above by a number depending only on $X$. Some of these results refer to the algebraic structure of the group, such as being abelian, or nilpotent, or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups, or to the action on cohomology.

We define the discrete degree of symmetry $disc-sym(X)$ of a closed $n$-manifold $X$ as the biggest $m\geq 0$ such that $X$ supports an effective action of $({\mathbf Z}/r)^m$ for arbitrarily big values of $r$. We prove that if $X$ is connected then $disc-sym(X)\leq 3n/2$. We propose the question of whether for every closed connected $n$-manifold $X$ the inequality $disc-sym(X)\leq n$ holds true, and whether the only closed connected $n$-manifold $X$ for which $disc-sym(X)=n$ is the torus $T^n$. We prove partial results providing evidence for an affirmative answer to this question.

Let Γ be a finite group acting on a Lie group G . We consider a class of group extensions 1 → G →Ĝ→Γ→ 1 defined by this action and a 2-cocycle of Γ with values in the centre of G . We establish and study a correspondence between Ĝ -bundles on a manifold and twisted Γ -equivariant bundles with structure group G on a suitable Galois Γ -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group Ĝ , since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Γ is the group of connected components of Ĝ .

We prove that for any closed smooth $4$-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G

If a finite p-group G acts continuously on a compact topological manifold M then, with some bound C depending on M alone, G has a subgroup H of index at most C such that the H-action on M has at most C stabilizer subgroups. This result plays a crucial role in the proof of a deep conjecture of Ghys.

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. This generalizes Simpson's results for GL(n,C) to arbitrary complex and real reductive Lie groups. Three interesting features are the relation between the parabolic degree and the Tits geometry of the boundary at infinity of the symmetric space, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant–Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal Met}_f$ with the following property. Let $g\in{\mathcal Met}_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi$ of $M$ preserving $\nabla^gf$ there exists some real number $t$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma\,\Phi_t^g(x)$, where $\Phi_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.

If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some $x\in X$ whose stabilizer $G_x\leq G$ satisfies $[G:G_x]\leq C$. We say that $X$ has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if $X$ is compact and possibly with boundary and has no odd cohomology then $(X,Diff(X))$ has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if $Z$ is a non necessarily compact smooth real affine variety, and $Z$ has no odd cohomology, then $(Z,Aut(Z))$ has the almost fixed point property, where $Aut(Z)$ is the group of algebraic automorphisms of $Z$ lifting the identity on $Spec\,{\mathbb{R}}$.

We prove that for any closed Lorentz 4-manifold ( M , g ) the isometry group Isom(M,g) is Jordan. Namely, there exists a constant C (depending on M and g ) such that any finite subgroup Γ≤Isom(M,g) has an abelian subgroup A≤Γ satisfying [Γ :A]≤ C .

Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

Let $(X,\omega)$ be a compact symplectic manifold of dimension $2n$ and let $Ham(X,\omega)$ be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant $C$, depending on $X$ but not on $\omega$, such that any finite subgroup $G\subset Ham(X,\omega)$ has an abelian subgroup $A\subseteq G$ satisfying $[G:A]\leq C$, and $A$ can be generated by $n$ elements or fewer. If $b_1(X)=0$ we prove an analogous statement for the entire group of symplectomorphisms of $(X,\omega)$. If $b_1(X)\neq 0$ we prove the existence of a constant $C'$ depending only on $X$ such that any finite subgroup $G\subset Symp(X,\omega)$ has a subgroup $N\subseteq G$ which is either abelian or $2$-step nilpotent and which satisfies $[G:N]\leq C'$. These results are deduced from the classification of the finite simple groups, the topological rigidity of hamiltonian loops, and the following theorem, which we prove in this paper. Let $E$ be a complex vector bundle over a compact, connected, smooth and oriented manifold $M$; suppose that the real rank of $E$ is equal to the dimension of $M$, and that $\langle e(E),[M]\rangle\neq 0$, where $e(E)$ is the Euler class of $E$; then there exists a constant $C$ such that, for any prime $p$ and any finite $p$-group $G$ acting on $E$ by vector bundle automorphisms preserving an almost complex structure on $M$, there is a subgroup $G_0\subseteq G$ satisfying $M^{G_0}\neq\emptyset$ and $[G:G_0]\leq C$.

A recent preprint of Csikós, Pyber and Szabó [CPS] proves that the diffeomorphism group of T 2 × S 2 is not Jordan. The purpose of this paper is to generalize the arguments in [CPS] in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any ∈ > 0 there exist compact manifolds admitting effective smooth actions of arbitrarily large finite groups Γ all of whose abelian subgroups have at most | Γ | ∈ elements. Finally, we also recover some results on the nonexistence of effective actions of compact connected semisimple Lie groups on manifolds.

For any symplectic form $\omega$ on $T^2\times S^2$ we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on $T^2\times S^2$ that are trivial in cohomology but which do not admit any effective symplectic action on $(T^2\times S^2,\omega)$. We also prove that for any $\omega$ there is another symplectic form $\omega'$ on $T^2\times S^2$ and a finite group acting symplectically and effectively on $(T^2\times S^2,\omega')$ which does not admit any effective symplectic action on $(T^2\times S^2,\omega)$. A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of $T^2\times S^2$. A group $G$ is Jordan if there exists a constant $C$ such that any finite subgroup $\Gamma$ of $G$ contains an abelian subgroup whose index in $\Gamma$ is at most $C$. Csikos, Pyber and Szabo proved recently that the diffeomorphism group of $T^2\times S^2$ is not Jordan. We prove that, in contrast, for any symplectic form $\omega$ on $T^2\times S^2$ the group of symplectomorphisms $Symp(T^2\times S^2,\omega)$ is Jordan. We also give upper and lower bounds for the optimal value of the constant $C$ in Jordan's property for $Symp(T^2\times S^2,\omega)$ depending on the cohomology class represented by $\omega$. Our bounds are sharp for a large class of symplectic forms on $T^2\times S^2$.

Let G be a compact connected semisimple Lie group with Lie algebra 𝔤. Let 𝒪⊂𝔤^* be a coadjoint orbit. The action of G on 𝒪 induces a morphism ρ:G→Homeo(𝒪). We prove that the induced map π_1(ρ):π_1(G)→π_1(Homeo(𝒪)) is injective. This strengthens a theorem of McDuff and Tolman (conjectured by Weinstein in 1989) according to which the analogous map G→Ham(𝒪) is injective on fundamental groups, where Ham(𝒪) is the group of Hamiltonian diffeomorphisms of the standard symplectic structure on 𝒪. To prove our theorem we associate to every nontrivial element of π_1(G) a bundle over S^2 with fiber 𝒪, using the standard patching construction. We then prove that the resulting bundle is topologically nontrivial by studying its cohomology. For this, we prove that it suffices to consider the case in which G is simple and 𝒪 is a minimal orbit, and then we prove our result for simple G and minimal orbit 𝒪 in a case by case analysis; the proof for the exceptional groups E_6 and E_7 relies on computer calculations, while all the other ones are addressed by hand. A basic tool in some of our computations is a generalization of Chevalley's formula to bundles of coadjoint orbits over S^2 that we prove in this paper.

We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ satisfying $[G:A]\leq C$, $\chi(X^A)=\chi(X)$, and $A$ can be generated by at most $2$ elements. Furthermore, if $\chi(X)<0$ then $A$ is cyclic. This proves, for any such $X$, a conjecture of Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.

Let C be a set of finite groups which is closed under taking subgroups and let d and M be positive integers. Suppose that for every G∈C whose order is divisible by at most two distinct primes there exists an abelian subgroup A⊆G such that A is generated by d or fewer elements and [G:A]≤M. We prove that there exists a positive constant C0 such that every G∈C has an abelian subgroup A satisfying [G:A]≤C0, and A can be generated by d or fewer elements. We also prove some related results. Our proofs use the Classification of Finite Simple Groups.

Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.