Mapping torus

of an automorphism of a manifold

The mapping torus of a self-mapping is the identification space

which is equipped with a canonical mapping

If is a closed -dimensional manifold and is an automorphism, then is a closed -dimensional manifold such that is the projection of a fibre bundle (cf. also Fibration) with fibre and monodromy . If is an -dimensional manifold with boundary and is an automorphism such that , then is an -dimensional manifold with boundary , and the union

is a closed -dimensional manifold, called an open book. It is important to know when manifolds are fibre bundles over and open books, for in those cases the classification of -dimensional manifolds is reduced to the classification of automorphisms of -dimensional manifolds.

A codimension- submanifold is fibred if it has a neighbourhood such that the exterior is a mapping torus, i.e. if is an open book for some automorphism of a codimension- submanifold with (a Seifert surface, cf. Seifert manifold). Fibred knots and fibred links have particularly strong geometric and algebraic properties (cf. also Knot and link diagrams; Knot theory).

In 1923, J.W. Alexander used geometry to prove that every closed -dimensional manifold is an open book, that is, there exists a fibred link , generalizing the Heegaard splitting.

Fibred knots came to prominence in the 1960s with the influential work of J. Milnor on singular points of complex hypersurfaces, and with the examples of E. Brieskorn realizing the exotic spheres as links of singular points.

Connected infinite cyclic coverings of a connected space are in one-one correspondence with expressions of the fundamental group as a group extension

and also with the homotopy classes of mappings inducing surjections . If is the projection of a fibre bundle with compact, the non-compact space is homotopy equivalent (cf. also Homotopy type) to the fibre , which is compact, so that the fundamental group and the homology groups are finitely generated.

In 1962, J. Stallings used group theory to prove that if is an irreducible closed -dimensional manifold with and with an extension such that is finitely generated (cf. also Finitely-generated group), then is a fibre bundle over , with for some automorphism of a surface . In 1964, W. Browder and J. Levine used simply-connected surgery to prove that for every closed -dimensional manifold with and finitely generated, is a fibre bundle over . In 1984, M. Kreck used this type of surgery to compute the bordism groups of automorphisms of high-dimensional manifolds and to evaluate the mapping-torus mapping

to the ordinary bordism over (cf. also Bordism).

A band is a compact manifold with a connected infinite cyclic covering which is finitely dominated, i.e. such that there exists a finite CW-complex with mappings , and a homotopy . In 1968, F.T. Farrell used non-simply-connected surgery theory to prove that for a piecewise-linear (or differentiable) -dimensional manifold band is a fibre bundle over if and only if a Whitehead torsion obstruction is . The theorem was important in the structure theory of high-dimensional topological manifolds, and in 1970 was extended to topological manifolds by L. Siebenmann. There is also a version for Hilbert cube manifolds, obtained in 1974 by T.A. Chapman and Siebenmann. The fibering obstruction for finite-dimensional measures the difference between the intrinsic simple homotopy type of given by a handle-body decomposition and the extrinsic simple homotopy type given by with a generating covering translation.

In 1972, H.E. Winkelnkemper used surgery to prove that for a simply-connected -dimensional manifold is an open book if and only if the signature of is . In 1977, T. Lawson used non-simply-connected surgery to prove that for odd every -dimensional manifold is an open book. In 1979, F. Quinn used non-simply-connected surgery to prove that for even an -dimensional manifold is an open book if and only if an obstruction in the asymmetric Witt group of vanishes, generalizing the Wall surgery obstruction (cf. also Witt decomposition).

For a recent account of fibre bundles over and open books see [a1].

References

[a1]	A. Ranicki, "High-dimensional knot theory" , Springer (1998)