MIXED STRUCTURES ON A MANIFOLD WITH BOUNDARY

Abstract

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 his classical book, Wall mentioned the possibility of introducing a mixed type of structure on a manifold with boundary. Following this suggestion, we introduce 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. The relationships can be described by using certain braids of exact sequences. Finally, we discuss explicitly several geometric examples.