Mapping torus

of an automorphism of a manifold

The mapping torus of a self-mapping $h : F \rightarrow F$ is the identification space

\begin{equation*} T ( h ) = F \times [ 0,1 ] / \{ ( x , 0 ) \sim ( h ( x ) , 1 ) : x \in F \}, \end{equation*}

which is equipped with a canonical mapping

\begin{equation*} p : T ( h ) \rightarrow S ^ { 1 } = [ 0,1 ] / \{ 0 \sim 1 \}, \end{equation*}

\begin{equation*} ( x , t ) \rightarrow t. \end{equation*}

If $F$ is a closed $n$-dimensional manifold and $h : F \rightarrow F$ is an automorphism, then $T ( h )$ is a closed $( n + 1 )$-dimensional manifold such that $p$ is the projection of a fibre bundle (cf. also Fibration) with fibre $F$ and monodromy $h$. If $F$ is an $n$-dimensional manifold with boundary and $h : F \rightarrow F$ is an automorphism such that $h | _ { \partial F } = 1 : \partial F \rightarrow \partial F$, then $T ( h )$ is an $( n + 1 )$-dimensional manifold with boundary $\partial T ( h ) = \partial F \times S ^ { 1 }$, and the union

\begin{equation*} t ( h ) = T ( h ) \bigcup_{ \partial T ( h )} \partial F \times D ^ { 2 } \end{equation*}

is a closed $( n + 1 )$-dimensional manifold, called an open book. It is important to know when manifolds are fibre bundles over $S ^ { 1 }$ and open books, for in those cases the classification of $( n + 1 )$-dimensional manifolds is reduced to the classification of automorphisms of $n$-dimensional manifolds.

A codimension-$2$ submanifold $K ^ { n } \subset M ^ { n + 2 }$ is fibred if it has a neighbourhood $K \times D ^ { 2 } \subset M$ such that the exterior $X = \operatorname { cl } ( M \backslash ( K \times D ^ { 2 } ) )$ is a mapping torus, i.e. if $X = t ( h )$ is an open book for some automorphism $h : F \rightarrow F$ of a codimension-$1$ submanifold $F ^ { n + 1 } \subset M$ with $\partial F = K$ (a Seifert surface, cf. Seifert manifold). Fibred knots $S ^ { n } \subset S ^ { n + 2 }$ and fibred links $\cup S ^ { n } \subset S ^ { n + 2 }$ 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 $3$-dimensional manifold $M ^ { 3 }$ is an open book, that is, there exists a fibred link $\cup S ^ { 1 } \subset M$, generalizing the Heegaard splitting.

Fibred knots $S ^ { n } \subset S ^ { n + 2 }$ 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 $\overline{M}$ of a connected space $M$ are in one-one correspondence with expressions of the fundamental group $\pi _ { 1 } ( M )$ as a group extension

\begin{equation*} \mathcal{E} : 1 \rightarrow \pi _ { 1 } ( \overline { M } ) \rightarrow \pi _ { 1 } ( M ) \rightarrow \mathbf{Z} \rightarrow \{ 1 \}, \end{equation*}

and also with the homotopy classes of mappings $p : M \rightarrow S ^ { 1 }$ inducing surjections $p * : \pi _ { 1 } ( M ) \rightarrow \pi _ { 1 } ( S ^ { 1 } ) = \mathbf{Z}$. If $p : M \rightarrow S ^ { 1 }$ is the projection of a fibre bundle with $M$ compact, the non-compact space $\overline{M}$ is homotopy equivalent (cf. also Homotopy type) to the fibre $F$, which is compact, so that the fundamental group $\pi _ { 1 } ( \overline { M } ) = \pi _ { 1 } ( F )$ and the homology groups $H_{ *} ( \overline { M } ) = H_{ *} ( F )$ are finitely generated.

In 1962, J. Stallings used group theory to prove that if $M$ is an irreducible closed $3$-dimensional manifold with $\pi _ { 1 } ( M ) \neq \mathbf{Z} _ { 2 }$ and with an extension $\cal E$ such that $\pi _ { 1 } ( \overline { M } )$ is finitely generated (cf. also Finitely-generated group), then $M$ is a fibre bundle over $S ^ { 1 }$, with $M = T ( h )$ for some automorphism $h : F \rightarrow F$ of a surface $F ^ { 2 } \subset M$. In 1964, W. Browder and J. Levine used simply-connected surgery to prove that for $n \geq 6$ every closed $n$-dimensional manifold $M$ with $\pi _ { 1 } ( M ) = \mathbf{Z}$ and $H_{ *} ( \overline { M } )$ finitely generated, is a fibre bundle over $S ^ { 1 }$. In 1984, M. Kreck used this type of surgery to compute the bordism groups $\Delta_*$ of automorphisms of high-dimensional manifolds and to evaluate the mapping-torus mapping

\begin{equation*} T : \Delta _ { n } \rightarrow \Omega _ { n + 1 } ( S ^ { 1 } ), \end{equation*}

\begin{equation*} ( F ^ { n } , h : F \rightarrow F ) \rightarrow T ( h ), \end{equation*}

to the ordinary bordism over $S ^ { 1 }$ (cf. also Bordism).

A band is a compact manifold $M$ with a connected infinite cyclic covering $\overline{M}$ which is finitely dominated, i.e. such that there exists a finite CW-complex $K$ with mappings $f : \overline { M } \rightarrow K$, $g : K \rightarrow \overline { M }$ and a homotopy $g f \simeq 1 : \overline { M } \rightarrow \overline { M }$. In 1968, F.T. Farrell used non-simply-connected surgery theory to prove that for $n \geq 6$ a piecewise-linear (or differentiable) $n$-dimensional manifold band $M$ is a fibre bundle over $S ^ { 1 }$ if and only if a Whitehead torsion obstruction $\Phi ( M ) \in \operatorname{Wh} ( \pi _ { 1 } ( M ) )$ is $0$. 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 $\Phi ( M )$ for finite-dimensional $M$ measures the difference between the intrinsic simple homotopy type of $M$ given by a handle-body decomposition and the extrinsic simple homotopy type given by $M \simeq T ( \zeta )$ with $\zeta : \overline { M } \rightarrow \overline { M }$ a generating covering translation.

In 1972, H.E. Winkelnkemper used surgery to prove that for $n \geq 7$ a simply-connected $n$-dimensional manifold $M$ is an open book if and only if the signature of $M$ is $0$. In 1977, T. Lawson used non-simply-connected surgery to prove that for odd $n \geq 7$ every $n$-dimensional manifold $M$ is an open book. In 1979, F. Quinn used non-simply-connected surgery to prove that for even $n \geq 6$ an $n$-dimensional manifold $M$ is an open book if and only if an obstruction in the asymmetric Witt group of $\mathbf{Z} [ \pi _ { 1 } ( M ) ]$ vanishes, generalizing the Wall surgery obstruction (cf. also Witt decomposition).

For a recent account of fibre bundles over $S ^ { 1 }$ and open books see [a1].

References

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