Mixed structures on a manifold with boundary

For a closed topological n-manifold X, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of X. In the case of a compact topological n-manifold with boundary (X, \partial X), the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In this paper we introduce new mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. Finally, we discuss several geometric examples.