Christopher Allday

On the rank of a space

The rank of a space is defined as the dimension of the highest dimensional torus which can act almost-freely on the space. (By an almost-free action is meant one for which all the isotropy subgroups are finite.) This definition is shown to extend the classical definition of the rank of a Lie group. A conjecture giving an upper bound for the rank of a space in terms of its rational homotopy is investigated.

Examples of circle actions on symplectic spaces

Many interesting results in the study of symplectic torus actions can be proved by purely cohomological methods. All one needs is a closed orientable topological 2nmanifold M (or, more generally, a reasonably pleasant topological space whose rational cohomology satisfies Poincare duality with formal dimension 2n), which is cohomologically symplectic (c-symplectic) in the sense that there is a class w ∈ H(M ;Q) such that w 6= 0. Sometimes one requires that M satisifes the Lefschetz condition that multiplication by wn−1 is an isomorphism H(M ;Q) → H2n−1(M ;Q). And an action of a torus T on M is said to be cohomologically Hamiltonian (c-Hamiltonian) if w ∈ Im[i∗ : H(MT ;Q) → H∗(M ;Q)], where MT is the Borel construction; and i : M →MT is the inclusion of the fibre in the fibre bundle MT → BT . Some examples of some results which can be proved easily by cohomological methods are the following.

Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property

We prove a Poincare‐Alexander‐Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and nonorientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen‐Macaulayness of relative equivariant cohomology modules arising from the orbit filtration. 55N91; 13C14, 57R91

Poincaré duality in P.A. Smith theory

Let G = S 1 , G = Z=p or more generally G be a nite p- group, where p is an odd prime. If G acts on a space whose cohomology ring fullls Poincar e duality (with appropriate coecients k), we prove a mod 4 congruence between the total Betti number of X G and a number which depends only on the k(G)-module structure of H (X;k). This improves the well known mod 2 congruences that hold for actions on general spaces.

On the rational homotopy Lie algebra of a fixed point set of a torus action

Let X be a simply connected topological space, and let Y*(X) be its rational homotopy Lie algebra. Suppose that a torus acts on X with fixed points, and suppose that F is a simply connected component of the fixed point set. If Y*( X) is finitely presented and if F is full, then it is shown that Y*(F) is finitely presented, and that the numbers of generators and relations in a minimal presentation of *( F) do not exceed the numbers of generators and relations (respectively) in a minimal presentation of Y*( X). Various other related results are given.