Homeomorphisms of 3-manifolds with compressible boundary.

*(English)*Zbl 0602.57011
Mem. Am. Math. Soc. 344, 100 p. (1986).

Let M be a compact, orientable, irreducible and sufficiently large 3- manifold and let H(M) be the mapping class group of M (i.e. the group of isotopy classes of homeomorphisms). A twist of M is a homeomorphism obtained by cutting M along a properly embedded disk, then twisting one of the two copies of the disk by \(2\pi\), and gluing the copies back together.

In the first third of the memoir it is shown (Theorems 4.2.1 and 4.2.2) that H(M) is finitely generated and that the subgroup of H(M) generated by twists and by Dehn twists about essential annuli and tori has finite index in H(M). If M is boundary-irreducible this was proved by K. Johannson [Homotopy equivalences of 3-manifolds with boundaries (Lect. Notes Math. 761) (1979; Zbl 0412.57007)]. If M has a compressible boundary component F, the authors construct an ”incompressible neighbourhood” V of F, which is unique up to ambient isotopy of M, and which is a ”product-with-handles” (i.e. a boundary-connected sum of a handlebody and trivial I-bundles over aspherical 2-manifolds). After exhibiting specific generators for H(V) and showing that H(V) is finitely generated, Theorems 4.2.1 and 4.2.2 are proved by induction on the number of compressible boundary components of M (starting the induction with the boundary-irreducible case).

In the second third of the memoir it is shown (Theorem 6.2.1) that a homeomorphism h of M that induces the identity on the fundamental group is isotopic to a product of twists if \(\deg (h)=1\). If \(\deg (h)=-1\) then M is an I-bundle over a 2-manifold. This generalizes a result proved by E. Luft for handlebodies [Math. Ann. 234, 279-292 (1978; Zbl 0364.57011)]. If M is boundary-irreducible this follows from the work of F. Waldhausen [Ann. Math., II. Ser. 87, 56-88 (1968; Zbl 0157.306)] and F. Laudenbach [Astérisque 12, 1-152 (1974; Zbl 0293.57004)]. In the more complicated general case the theorem is proved by induction on the number of compressible boundary components F of M, after it has been proved for the product-with-handles neighbourhood of F.

In the third part of the memoir the results are generalized to non- orientable \(P^ 2\)-irreducible 3-manifolds.

In the first third of the memoir it is shown (Theorems 4.2.1 and 4.2.2) that H(M) is finitely generated and that the subgroup of H(M) generated by twists and by Dehn twists about essential annuli and tori has finite index in H(M). If M is boundary-irreducible this was proved by K. Johannson [Homotopy equivalences of 3-manifolds with boundaries (Lect. Notes Math. 761) (1979; Zbl 0412.57007)]. If M has a compressible boundary component F, the authors construct an ”incompressible neighbourhood” V of F, which is unique up to ambient isotopy of M, and which is a ”product-with-handles” (i.e. a boundary-connected sum of a handlebody and trivial I-bundles over aspherical 2-manifolds). After exhibiting specific generators for H(V) and showing that H(V) is finitely generated, Theorems 4.2.1 and 4.2.2 are proved by induction on the number of compressible boundary components of M (starting the induction with the boundary-irreducible case).

In the second third of the memoir it is shown (Theorem 6.2.1) that a homeomorphism h of M that induces the identity on the fundamental group is isotopic to a product of twists if \(\deg (h)=1\). If \(\deg (h)=-1\) then M is an I-bundle over a 2-manifold. This generalizes a result proved by E. Luft for handlebodies [Math. Ann. 234, 279-292 (1978; Zbl 0364.57011)]. If M is boundary-irreducible this follows from the work of F. Waldhausen [Ann. Math., II. Ser. 87, 56-88 (1968; Zbl 0157.306)] and F. Laudenbach [Astérisque 12, 1-152 (1974; Zbl 0293.57004)]. In the more complicated general case the theorem is proved by induction on the number of compressible boundary components F of M, after it has been proved for the product-with-handles neighbourhood of F.

In the third part of the memoir the results are generalized to non- orientable \(P^ 2\)-irreducible 3-manifolds.

Reviewer: W.Heil

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57R50 | Differential topological aspects of diffeomorphisms |