Fixed sets of involutions

Fixed Point Sets of Involutions on Spheres

The Mod 2 Cohomology of Fixed Point Sets of Anti-symplectic Involutions

Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus G = Tn. Suppose that σ is an anti-symplectic involution compatible with the G-action. The real locus of M is X, the fixed point set of σ. Duistermaat uses Morse theory to give a description of the ordinary cohomology of X in terms of the cohomology of M . There is a residual GR = (Z/2Z) ...

Geometric structures associated to triangulations as fixed point sets of involutions

Thurston’s first examples of hyperbolic structures on the complement of links were obtained by using topological ideal triangulations, that is, triangulations with removed vertices [T]. That idea was the source of a huge number of examples and the proof of the hyperbolic Dehn surgery theorem. Recently, examples of spherical CR structures (locally modelled on the Heisenberg group, see section 5)...

Fixed Point Shifts of Inert Involutions

Given a mixing shift of finite type X , we consider which subshifts of finite type Y ⊂ X can be realized as the fixed point shift of an inert involution of X . We establish a condition on the periodic points of X and Y that is necessary for Y to be the fixed point shift of an inert involution of X . We show that this condition is sufficient to realize Y as the fixed point shift of an involution...

Fixed point free involutions on Riemann surfaces

Involutions without fixed points on hyperbolic closed Riemann surface are discussed. Notably, the following question is answered: for τ such an involution on a surface (S, d), can a point p be chosen so that d(p, τ(p)) is bounded by a constant dependent solely on the genus of S? This is proved to be true for even genus and false otherwise. Furthermore, the sharp constant is given in genus 2.