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Session 31. Representation Theory, Transformation Groups, and Applications
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Smooth Group Actions on Homotopy Complex Projective Spaces |
Marek Kaluba, Adam Mickiewicz University in Poznań, Poland
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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.
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References- M. Kaluba, Constructions of smooth exotic actions on homotopy
complex projective spaces and products of manifolds , PhD Thesis, UAM
Poznań, March 2014.
- M. Morimoto and K. Pawałowski, Smooth actions of finite
Oliver groups on spheres , Topology 42, Issue 2 (2003), pp. 395-421.
- M. Morimoto, Fixed-point sets of smooth actions on spheres ,
Journal of \(K\)-theory 1, Issue 01 (2008), pp. 95-128.
- B. Oliver, Fixed point sets and tangent bundles of actions
on disks and Euclidean spaces , Topology 35, Issue 3 (1996), pp. 583-615.
- 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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