Mapping torus

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In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism:

M_f =\frac{(X \times I)}{(x,0)\sim (f(x),1)}

The result is a fiber bundle whose base is a circle and whose fiber is the original space Y.

If X is a manifold, Mf will be a manifold of dimension one grander, and it is said to "fiber over the circle". 3-manifolds which fiber through the circle have been modestly studied.