Krzysztof Pawałowski

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2-proper action on a sphere with isolated fixed points at which the tangent representations of G are mutually nonisomorphic if and only if G contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer n > 1 and primes p, q, we prove similar results for the group G = AnX SL2 (Fp), or PSL2 (Fq). In particular, G has Smith equivalent representations that are not isomorphic if and only if n > 8, p > 5, q > 19.

Smooth actions of finite Oliver groups on spheres

Abstract In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S , which smooth manifolds F occur as the fixed point sets in S , and which real G -vector bundles ν over F occur as the equivariant normal bundles of F in S ? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G , F , and ν . We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Olivers equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.

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

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.

The Laitinen Conjecture for finite solvable Oliver groups

For smooth actions of G on spheres with exactly two fixed points, the Laitinen Conjecture proposed an answer to the Smith question about the G-modules determined on the tangent spaces at the two fixed points. Morimoto obtained the first counterexample to the Laitinen Conjecture for G = Aut(A 6 ). By answering the Smith question for some finite solvable Oliver groups G, we obtain new counterexamples to the Laitinen Conjecture, presented for the first time in the case where G is solvable.

NON-SYMPLECTIC SMOOTH CIRCLE ACTIONS ON SYMPLECTIC MANIFOLDS

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.

Corrections to: Fixed point sets of smooth group actions on disks and Euclidean spaces

THE GOAL of this note is to correct statements of some assertions in [4]. The author would like to thank Agnieszka Bojanowska and Bob Oliver for pointing out a mistake in the proof of Proposition 4.4 in [4]. The mistake occurs in the G-vector bundle extension arguments. We note that Proposition 4.4 is not true due to a counterexample provided by Bob Oliver. We wish to mention all assertions in [4] which have used Proposition 4.4 with the incorrect statement. First of all, in the proof of Theorem 4.5 in [4], instead of using Proposition 4.4 in [4], we may apply Proposition 4.4 from [3] to complete the arguments in the proof of Theorem 4.5 in [4]. Proposition 5.5 in [4] should be corrected as follows. For a prime p and an integer n 2 1, denote by Sin-r the sphere S’“-l with the standard free action of the cyclic group of order p. For distinct primes pl, p2, . . . ,p& and any finite dimensional CW complex M, the