scholarly journals Non-symplectic smooth circle actions on symplectic manifolds

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Bogusław Hajduk ◽  
Krzysztof Pawałowski ◽  
Aleksy Tralle
Keyword(s):  
Fixed Point ◽  
Symplectic Form ◽  
Symplectic Manifolds ◽  
Point Sets ◽  
Mapping Torus ◽  
Connected Sum ◽  
Circle Actions ◽  
Periodic Diffeomorphisms ◽  
Carry Over ◽  
Fixed Point Sets

AbstractWe 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.

scholarly journals Rational homotopy of the (homotopy) fixed point sets of circle actions

2009 ◽  
Vol 222 (1) ◽  
pp. 151-171 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo
Keyword(s):  
Fixed Point ◽  
Rational Homotopy ◽  
Point Sets ◽  
Circle Actions ◽  
Fixed Point Sets

Fixed point sets for abstract volterra operators on fréchet spaces

2000 ◽  
Vol 76 (1-2) ◽  
pp. 131-152 ◽  
Author(s):  
Dana M. Bedivan ◽  
Donal O′Regan
Keyword(s):  
Fixed Point ◽  
Fréchet Spaces ◽  
Point Sets ◽  
Volterra Operators ◽  
Fixed Point Sets

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

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