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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?
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?