Conjugations on 6-manifolds with free integral cohomology

In this article, we show the existence of conjugations on many simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X^6 to admit a conjugation with fixed point set M^3 is the obvious one: the existence of a degree-halving ring isomorphism between the Z_2-cohomologies of X and M.Comment: 23 page

Conjugations on 6-Manifolds

Conjugation spaces are spaces with involution such that the fixed point set of the involution has Z/2-cohomology isomorphic to the Z/2-cohomology of the space itself, with the little difference that all degrees are divided by two (e.g. CP^n with the complex conjugation). One also requires that a certain conjugation equation is fulfilled. I give a new characterization of conjugation spaces and apply it to the following realization question: given M, a closed orientable 3-manifold, is there a 6-manifold X (with certain additional properties) containing M as submanifold such that M is the fixed point set of an orientation reversing involution on X? My main result is that for every such 3-manifold M there exists a simply connected conjugation 6-manifold X with fixed point set M

Involutions on S^6 with 3-dimensional fixed point set

In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.Comment: 23 page

One-connectivity and finiteness of Hamiltonian S1S^1-manifolds with minimal fixed sets

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology