ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY
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Transformation Groups
ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY IGNASI MUNDET I RIERA∗ Facultat de Matem`atiques i Inform`atica Universitat de Barcelona Gran Via de les Corts Catalanes 585 08007 Barcelona, Spain [email protected]
Abstract. If X is a smooth manifold and G is a subgroup of Diff(X) we say that (X, G) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ G there is some x ∈ X whose stabilizer Gx ≤ G satisfies [G : Gx ] ≤ 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 R. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λstability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.
1. Introduction 1.1. Smooth actions Let X be a smooth manifold, possibly with boundary, and let G be a subgroup of Diff(X). We say that (X, G) has the fixed point property if for any finite subgroup G ≤ G the fixed point set X G is nonempty. It is natural to ask for which manifolds X does the pair (X, Diff(X)) have the fixed point property. This is trivially the case for asymmetric manifolds i.e., manifolds which do not admit an effective action of any nontrivial finite group (there exist many examples of asymmetric manifolds, see, e.g., [3], [8], [30]). The DOI: 10.1007/S00031-019-09534-7 This work has been partially supported by the (Spanish) MEC Projects MTM201238122-C03-02 and MTM2015-65361-P. Received July 19, 2018. Accepted December 6, 2018. Corresponding Author: Ignasi Mundet i Riera, e-mail: [email protected] ∗
IGNASI MUNDET I RIERA
question is more interesting if one further requires that Diff(X) contains nontrivial finite groups, preferably of arbitrarily big size. If Dn denotes the closed n-dimensional disk, then (D n , Diff(Dn )) has the fixed point property if n ≤ 4 (this is an easy exercise for n ≤ 2, and is much less obvious if n is 3 or 4, see [6]). However, (Dn , Diff(Dn )) does not have the fixed point property if n ≥ 6: for any n ≥ 6 there is a smooth effective action of the alternating group A5 on Dn (see [1], and note that a one-fixed-point action on S n induces a fixed point free action on Dn by removing an invariant open ball in S n centered at the fixed point). Similarly (Rn , Diff(Rn )) has the fixed point property if n ≤ 3 and it does not have it if n ≥ 5, see [6]. Given the previous results, it seems reasonable to expect that there are very few non-asymmetric manifolds X such that (X, D
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