Session 31. Representation Theory, Transformation Groups, and Applications

Smooth Group Actions on Homotopy Complex Projective Spaces

Marek Kaluba, Adam Mickiewicz University in Poznań, Poland
The fixed point sets of smooth finite group actions on Euclidean spaces, disks and spheres have been studied extensively. In particular, a complete description of the fixed point sets of such actions on disks has been obtained in [5] and [4]. In the case of actions on spheres, a similar characterisation has been found for finite perfect groups (see [2] and [3]). In this talk, we present a description of the fixed point sets of smooth actions of finite perfect groups on manifolds homotopy equivalent to complex projective spaces. In [1], we show that the following theorem holds.

Theorem: Let \(G\) be a finite perfect group with an element not of prime power order. Let \(F\) be a closed smooth manifold with an even dimensional connected component. If \(F\) is diffeomorphic to the fixed point set of a smooth action of \(G\) on a sphere, then \(F\) can be realised as the fixed point set of a smooth action of \(G\) on a complex projective space.

In particular, if \(G\) has a \(2n\)-dihedral sub-quotient for a composite natural number \(n\), then any closed smooth manifold is diffeomorphic to the fixed point set of a smooth action of \(G\) on a complex projective space. If \(G=A_5\) and \(F\) is a closed smooth manifold such that \([\tau_F] \in \text{Tor} \, \widetilde{KO}(F)\) and all connected components of \(F\) are of the same even dimension, we show that \(F\) can be realised as the fixed point set of a smooth action of \(G\) on a closed smooth manifold homotopy equivalent to a complex projective space.

References
  1. M. Kaluba, Constructions of smooth exotic actions on homotopy complex projective spaces and products of manifolds , PhD Thesis, UAM Poznań, March 2014.
  2. M. Morimoto and K. Pawałowski, Smooth actions of finite Oliver groups on spheres , Topology 42, Issue 2 (2003), pp. 395-421.
  3. M. Morimoto, Fixed-point sets of smooth actions on spheres , Journal of \(K\)-theory 1, Issue 01 (2008), pp. 95-128.
  4. B. Oliver, Fixed point sets and tangent bundles of actions on disks and Euclidean spaces , Topology 35, Issue 3 (1996), pp. 583-615.
  5. K. Pawałowski, Fixed point sets of smooth group actions on disks and Euclidean spaces , Topology 28, Issue 3 (1989), pp. 273-289.
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