ABSTRACT
The theory of handles in manifolds with group actions is more complex than it is in the inequivariant case. However, complications arise at a more fundamental level because the sweeping existence theorems for inequivariant handle decompositions fail dramatically, even for locally smoothable actions of finite groups, and when they exist, torsion invariants computed from them are usually not topological invariants. Handle decompositions have played a dominant role in the development of the theory of manifolds by both geometric and algebraic techniques since they exhibit the manifolds as unions of basic units assembled in relatively nice ways that capture the essential combinatorial aspects of their homological structure. Morse theory provides smooth handle decompositions for smooth manifolds, and for piecewise linear manifolds the stars of the barycenters of a given combinatorial triangulation with respect to its second barycentric subdivision form a PL-handle decomposition.
