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Assignments and abstract moment maps. (English) Zbl 0977.53078

Summary: Abstract moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originated in the study of cobordisms of Hamiltonian group actions.
In this paper we answer the question of existence of a (proper) abstract moment map for a torus action and give a necessary and sufficient condition for an abstract moment map to be associated with a pre-symplectic form. This is done by using the notion of an assignment, which is a combinatorial counterpart of an abstract moment map.
Finally, we show that the space of assignments fits as the zeroth cohomology in a series of certain cohomology spaces associated with a torus action on a manifold. We study the resulting “assignment cohomology” theory.

MSC:

53D20 Momentum maps; symplectic reduction
57R85 Equivariant cobordism
57S25 Groups acting on specific manifolds
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