Classification of Hamiltonian torus actions 

 

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                

 

 

 

 

 

(For precise definitions, statements and proofs see the paper

"New tools for classifying Hamiltonian circle actions with isolated fixed points" by L. Godinho and myself and the references therein.)

 

Given a compact manifold M and a Lie group G, it is usually hard to determine whether M admits a G action, i.e. to determine whether there exists a (non-trivial) homomorphism from G to the diffeomorphism group of M.   

 

A related question was posed by Ted Petrie in the seventies: given a compact manifold M homotopically equivalent to the complex projective space of the same dimension, does the existence of a non-trivial circle action imply that the Pontrjagin classes agree with those of the complex projective space? This question has been answered in many particular cases, but the general case remains unsolved. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                  

         

 

 

 

 

 

 

Assume that G is a compact real torus.

 

One way of knowing whether a manifold M admits a G action is given by studying the implications of the

existence of a G action at the level of the topological invariants of M, for example the cohomology ring and Chern classes. 

For certain "nice spaces" (see below) these can be computed from the equivariant counterparts which, in turns, can be computed from the isotropy representation of G on the tangent space restricted to the fixed point set. 

 

What I mean by a "nice space" is a compact symplectic manifold M with a Hamiltonian G action and isolated fixed points such that either

- the Betti numbers agree with those of the complex projective space of the same dimension, or

- the pair (M,G) is a GKM space.

 

Motivated by the above discussion, we pose the following:

 

Question: 
Given a compact symplectic manifold M of dimension 2n and Betti numbers b0, .... , b{2n}, with a Hamiltonian G action and isolated fixed points, can we classify all the possible representations of G on the tangent space at the fixed point set?

 

In the paper:

 

New tools for classifying Hamiltonian circle actions with isolated fixed points

 

L. Godinho and I introduce a new approach for answering this question. Namely, we prove the existence

of an invariant of the action which only depends on the Betti numbers of M, and that allows us to make

this problem solvable by using a combination of Mathematica and C++. 

The accompanying software can be found here.

In particular, we solve the aforementioned classification problem for a special class of eight-dimensional compact symplectic manifolds (see Theorem 1.3 and Corollary 1.4).

 

 

 

                                                                                                         

 

 

 

 


  

 

 




 

" There are two ways to do great mathematics. The first way is to be smarter than everybody else. The second way is to be stupider than everybody else, but persistent."  - Raoul Bott                                                                                                                                                                            

The standard S^1 action on the sphere: it is a Hamiltonian S^1 action with isolated fixed points

I would like to thank S. Tolman for introducing me to this problem, and L. Godinho for being the most optimistic and energetic collaborator ever.

Prof. SILVIA SABATINI

Ph.D.