Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we can add conditions to h like, say, respect a possible PL structure, or for h to be a diffeomorphism, etc.) , then we do the quotient on X x I ; I=[0,1]: X/~ : (x,0)~ (h(x),1 ) i.e., we glue the top- and bottom levels about the homeomorphism h. In the boundaryless case, the possible trouble points are those that are identified, since XxI is itself a manifold. Some trivial examples-- h is the identity -- are manifolds, but I cannot see clearly the case for general h. Say (x,0)~(x',1) . Then there are charts for x, x' respectively in X: For x: (Ux, Phi_x) , and (Ux', Phi_x' ) for x' How do we get a chart for the identified point (x,0)~(x',1) in X/~ ? The case where X has boundary I suspect, the identification process has the effect of capping the boundary. If X has boundary B, then h: X-->X takes B to B. But I can't think of how to make this more rigorous. Any ideas?