Mapping torus
of an automorphism of a manifold
The mapping torus of a self-mapping is the identification space
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which is equipped with a canonical mapping
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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
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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
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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
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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) |
Mapping torus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping_torus&oldid=49927