# Article

Ufa Mathematical Journal
Volume 8, Number 3, pp. 122-129

# Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary

Savin A.Yu., Sternin B.Yu.

DOI:10.13108/2016-8-3-122

Given an action of a discrete group $G$ on a smooth compact manifold $M$ with a boundary, we consider a class of operators generated by pseudodifferential operators on $M$ and shift operators associated with the group action. For elliptic operators in this class, we obtain a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the $K$-group of the crossed product of the algebra of continuous functions on the cotangent bundle over the interior of the manifold and the group $G$ acting on this algebra by automorphisms.