#### WWGD

Science Advisor

Gold Member

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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?