Complete classification of equivariant SK-invariants of G-manifolds

Determine, for an arbitrary finite group G, which equivariant cut-and-paste (SK) invariants completely classify equivariant cut-and-paste manifolds (i.e., G-manifolds equipped with a smooth G-action) up to equivariant SK-equivalence, extending the cases where a complete classification is already known.

Background

Cut-and-paste (SK) equivalence asks whether one manifold can be cut into pieces and reassembled to form another, with invariants such as the Euler characteristic classifying equivalence in the non-equivariant setting. In the classical case, Karras–Kreck–Neumann–Ossa showed that SK-equivalence classes are completely determined by algebraic invariants (Euler characteristic and signature in dimension 4).

In the equivariant setting, SK-invariants are studied for manifolds with a group action. The equivariant Euler characteristic remains an SK-invariant, but it is not complete. While complete classifications exist for certain groups (e.g., finite Abelian groups of odd order), identifying a full set of invariants that classify equivariant cut-and-paste manifolds for general G remains unresolved. This paper constructs a genuine G-spectrum for the squares K-theory of equivariant SK-manifolds, which organizes SK-invariants into Mackey functors and provides a framework that may aid in addressing the classification problem.