Advanced Search

We show that the property of having only vanishing triple Massey products in equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular it can be viewed as a partial answer to a question posed by Allday and Puppe about finding conditions ensuring the "formality" of G-actions.

In this note all vectors and $\epsilon$-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)={(detA)}^{\lambda }·A·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u.$

We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.