We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

We study the topological 4-dimensional surgery problem for a closed connected orientable topological 4-manifold $X$ with vanishing second homotopy and ${{\pi }_{1}}\left( X \right)\,\cong \,A\,*\,F\left( r \right)$ , where $A$ has one end and $F\left( r \right)$ is the free group of rank $r\,\ge \,1$ . Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.

We study the homotopy type and the s—cobordism class of a closed connected topological 4-manifold with vanishing second homotopy group. Our results are related to problem 4.53 of Kirby in Geometric Topology, Studies in Advanced Math. 2 (1997), and give a partial answer to a question stated by Hillman in Bull. London Math. Soc.27 (1995) 387–391.

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.

We study the s-cobordism type of closed orientable (smooth or PL) 4–manifolds with free or surface fundamental groups. We prove stable classification theorems for these classes of manifolds by using surgery theory.