##
**Cobordism of automorphisms of surfaces.**
*(English)*
Zbl 0535.57016

The paper is devoted to the computation of the cobordism group of diffeomorhisms of oriented surfaces. For \(n\geq 4\) the group \(\Delta_{n+}\) of cobordism classes of orientation preserving diffeomorphisms between orientable n-manifolds has been computed by M. Kreck[Topology 15, 353-361 (1976; Zbl 0335.57021); Bull Am. Math. Soc. 82, 759-761 (1976; Zbl 0329.57014)], and the group \(\Delta_{3+}\) has been computed by P. Melvin [Topology 18, 173-175 (1979; Zbl 0418.57014)]. Their methods give some partial information on \(\Delta_{2+}\), but A. Casson and independently K. Johannson and D. Johnson have shown that these methods are insufficient to determine \(\Delta_{2+}\) (unpublished). It turns out that for this problem the most difficult and interesting case is the two-dimensional one. (Of course, the problem of computation of \(\Delta_{2+}\) is not a purely two-dimensional problem, it is rather a three-dimensional problem.)

The main result of the paper is the following: \[ \Delta_{2+}\cong {\mathbb{Z}}^{\infty}\oplus({\mathbb{Z}}/2)^{\infty}\quad \Delta_ 2\cong {\mathbb{Z}}^{\infty}\oplus({\mathbb{Z}}/2)^{\infty}. \] where \(A^{\infty}\) denotes the direct sum of countably many copies of A. In fact, in the paper some more information on \(\Delta_ 2\) is obtained. Let \(\Delta^ p\!_ 2\) denote the group of periodic diffeomorphisms of surfaces modulo cobordism by periodic diffeomorphisms of 3-manifolds. It is proved in the paper that the canonical map \(\Delta^ p\!_ 2\to \Delta_ 2\) is injective. Also, the author introduces a set A of pseudo-Anosov diffeomorphisms (the set of diffeomorphisms which are minimal in a suitable sense) such that the canonical map \(A\to \Delta_ 2\) is injective and its image is a subgroup of \(\Delta_ 2\) (consequently, we can consider A as a group). It turns out that the canonical map \(\Delta^ p\!_ 2\times A\to \Delta_ 2\) is an isomorphism and both groups \(\Delta^ p\!_ 2\) and A are isomorphic to \({\mathbb{Z}}^{\infty}\times({\mathbb{Z}}/2)^{\infty}.\)

The proofs are based on several tools provided by the geomeric theory of 3-manifolds: the Johannson-Jaco-Shalen characteristic submanifold theorem, the Thurston’s hyperbolization theorem, the (3-dimensional case of) Mostow’s rigidity theorem etc. The main idea of the proofs is roughly the following. Let a surface diffeomorphism f: \(F\to F\) bound a null- cobordism F: \(M\to M\). If M is irredubible and \(\sigma\)-irreducible, then the Johannson-Jaco-Shalen characteristic submanifold of M is invariant (up to isotopy) under \(\hat f\) and so we can split (M,\^f) into several simpler pieces. Some pieces are hyperbolic. Here Mostow’s rigidity theorem applies, according to which every diffeomorphism of a hyperbolic manifold is periodic up to isotopy (this is the key point of the proof of injectivity \(\Delta^ p\!_ 2\to \Delta_ 2)\). Other pieces are still simpler (they are Seifert fibre spaces) and can be completely analyzed. The general case reduces to the irreducible case by an argument of M. Scharlemann, reproduced in an appendix, and then to the irreducible and \(\sigma\)-irreducible one by a theory of some new characteristic submanifolds developed in the paper.

Although the group \(\Delta_ 2\) is completely computed, the following problem remains in general unsettled: given an automorphism of a surface, decide whether it is null-cobordand or not. It is easy to see that it is sufficient to consider only irreducible diffeomorphisms. For periodic diffeomorphisms a good solution of this problem is obtained in the course of computation of \(\Delta^ p\!_ 2\). So the main difficulty here is in the pseudo-Anosov case.

In the last section of the paper some relations between this problem and the problem of extension of a surface diffeomorphism to a handlebody are presented, especially in the genus two case. The last problem appears to be easier to handle.

Similar results on \(\Delta_ 2\) were obtained by A. L. Edmonds and J. H. Ewing [Math. Ann. 259, 497-504 (1982; Zbl 0468.57023)]. Their methods are more algebraical in nature. In particular, they use the G- signature theorem instead of Thurston’s hyperbolization theorem to prove the injectivity of \(\Delta^ p\!_ 2\to \Delta_ 2\).

The main result of the paper is the following: \[ \Delta_{2+}\cong {\mathbb{Z}}^{\infty}\oplus({\mathbb{Z}}/2)^{\infty}\quad \Delta_ 2\cong {\mathbb{Z}}^{\infty}\oplus({\mathbb{Z}}/2)^{\infty}. \] where \(A^{\infty}\) denotes the direct sum of countably many copies of A. In fact, in the paper some more information on \(\Delta_ 2\) is obtained. Let \(\Delta^ p\!_ 2\) denote the group of periodic diffeomorphisms of surfaces modulo cobordism by periodic diffeomorphisms of 3-manifolds. It is proved in the paper that the canonical map \(\Delta^ p\!_ 2\to \Delta_ 2\) is injective. Also, the author introduces a set A of pseudo-Anosov diffeomorphisms (the set of diffeomorphisms which are minimal in a suitable sense) such that the canonical map \(A\to \Delta_ 2\) is injective and its image is a subgroup of \(\Delta_ 2\) (consequently, we can consider A as a group). It turns out that the canonical map \(\Delta^ p\!_ 2\times A\to \Delta_ 2\) is an isomorphism and both groups \(\Delta^ p\!_ 2\) and A are isomorphic to \({\mathbb{Z}}^{\infty}\times({\mathbb{Z}}/2)^{\infty}.\)

The proofs are based on several tools provided by the geomeric theory of 3-manifolds: the Johannson-Jaco-Shalen characteristic submanifold theorem, the Thurston’s hyperbolization theorem, the (3-dimensional case of) Mostow’s rigidity theorem etc. The main idea of the proofs is roughly the following. Let a surface diffeomorphism f: \(F\to F\) bound a null- cobordism F: \(M\to M\). If M is irredubible and \(\sigma\)-irreducible, then the Johannson-Jaco-Shalen characteristic submanifold of M is invariant (up to isotopy) under \(\hat f\) and so we can split (M,\^f) into several simpler pieces. Some pieces are hyperbolic. Here Mostow’s rigidity theorem applies, according to which every diffeomorphism of a hyperbolic manifold is periodic up to isotopy (this is the key point of the proof of injectivity \(\Delta^ p\!_ 2\to \Delta_ 2)\). Other pieces are still simpler (they are Seifert fibre spaces) and can be completely analyzed. The general case reduces to the irreducible case by an argument of M. Scharlemann, reproduced in an appendix, and then to the irreducible and \(\sigma\)-irreducible one by a theory of some new characteristic submanifolds developed in the paper.

Although the group \(\Delta_ 2\) is completely computed, the following problem remains in general unsettled: given an automorphism of a surface, decide whether it is null-cobordand or not. It is easy to see that it is sufficient to consider only irreducible diffeomorphisms. For periodic diffeomorphisms a good solution of this problem is obtained in the course of computation of \(\Delta^ p\!_ 2\). So the main difficulty here is in the pseudo-Anosov case.

In the last section of the paper some relations between this problem and the problem of extension of a surface diffeomorphism to a handlebody are presented, especially in the genus two case. The last problem appears to be easier to handle.

Similar results on \(\Delta_ 2\) were obtained by A. L. Edmonds and J. H. Ewing [Math. Ann. 259, 497-504 (1982; Zbl 0468.57023)]. Their methods are more algebraical in nature. In particular, they use the G- signature theorem instead of Thurston’s hyperbolization theorem to prove the injectivity of \(\Delta^ p\!_ 2\to \Delta_ 2\).

Reviewer: N.Ivanov

### MathOverflow Questions:

Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms### MSC:

57R50 | Differential topological aspects of diffeomorphisms |

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

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

37-XX | Dynamical systems and ergodic theory |

57R30 | Foliations in differential topology; geometric theory |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

### Keywords:

cobordism group of diffeomorhisms of oriented surfaces; periodic diffeomorphisms of surfaces; pseudo-Anosov diffeomorphisms; characteristic submanifold; hyperbolization; Mostow’s rigidity theorem; Seifert fibre spaces; extension of a surface diffeomorphism to a handlebody
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\textit{F. Bonahon}, Ann. Sci. Éc. Norm. Supér. (4) 16, 237--270 (1983; Zbl 0535.57016)

### References:

[1] | F. BONAHON , Cobordisme des difféomorphismes de surfaces (C. R. Acad. Sc., Paris, T. 290, série A, 1980 , pp. 765-767). MR 81d:57028 | Zbl 0464.57017 · Zbl 0464.57017 |

[2] | F. BONAHON and L. C. SIEBENMANN , New Geometric Splittings for Knots and Links , to appear. |

[3] | A. CASSON , Cobordism Invariants of Automorphisms of Surfaces , Handwritten Notes, Orsay, 1979 . |

[4] | A. EDMONDS and J. EWING , Remarks on the Cobordism Group of Surface Diffeomorphisms (Math. Ann., Vol. 259, 1982 , pp. 497-504). MR 83m:57026 | Zbl 0468.57023 · Zbl 0468.57023 |

[5] | A. FATHI and F. LAUDENBACH , Difféomorphismes pseudo-Anosov et décompositions de Heegaard (C. R. Acad. Sc., Paris, T. 291, série A, 1980 , pp. 423-425). MR 81k:57031 | Zbl 0459.57007 · Zbl 0459.57007 |

[6] | A. FATHI , F. LAUDENBACH and V, POENARU , Travaux de Thurston sur les surfaces (Astérisque, n^\circ 66-67, 1979 ). MR 82m:57003 |

[7] | G. HARDY and E. WRIGHT , An Introduction to the Theory of Numbers , Oxford University Press, 1938 . Zbl 0020.29201 | JFM 64.0093.03 · Zbl 0020.29201 |

[8] | W. JACO and P. SHALEN , Seifert Fibered Spaces in 3-Manifolds (Memoirs A.M.S., n^\circ 220, 1979 ). Zbl 0415.57005 · Zbl 0415.57005 |

[9] | J. JOHANNSON , Homotopy Equivalences of 3-Manifolds with Boundaries (Springer Lecture Notes, n^\circ 761, 1979 ). MR 82c:57005 | Zbl 0412.57007 · Zbl 0412.57007 |

[10] | J. JOHANNSON and D. JOHNSON , Non-Bounding Surface Diffeomorphisms which Act Trivially on the Homology , preprint, 1980 . |

[11] | H. KNESER , Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten (Jber. Deutsch. Math. Verein., Vol. 38, 1929 , pp. 248-260). Article | JFM 55.0311.03 · JFM 55.0311.03 |

[12] | M. KRECK , Cobordism of Odd Dimensional Diffeomorphisms (Topology, vol. 15, 1976 , pp. 353-361). MR 54 #11361 | Zbl 0335.57021 · Zbl 0335.57021 |

[13] | M. KRECK , Bordism of Diffeomorphisms (Bull. Amer. Math. Soc., vol. 82, 1976 , pp. 759-761). Article | MR 53 #11634 | Zbl 0329.57014 · Zbl 0329.57014 |

[14] | S. LOPEZ DE MEDRANO , Cobordism of Diffeomorphisms of (k - 1)-Connected 2k-Manifolds, Second Conference on Compact Transformations Groups , 217-227, Amherst, Springer, 1972 . MR 51 #1847 | Zbl 0259.57015 · Zbl 0259.57015 |

[15] | W. MEEKS and J. PATRUSKY , Representing Homology Classes by Embedded Circles on a Compact Surface (Illinois J. Math., Vol. 22, 1978 , pp. 262-269). Article | MR 57 #13951 | Zbl 0392.57016 · Zbl 0392.57016 |

[16] | W. MEEKS and S.-T. YAU , Topology of Three Dimensional Manifolds and the Embedding Problems in Minimal Surface Theory (Ann. Math., Vol. 112, 1980 , pp. 441-484). MR 83d:53045 | Zbl 0458.57007 · Zbl 0458.57007 |

[17] | P. MELVIN , Bordism of Diffeomorphisms (Topology, Vol. 18, 1979 , pp. 173-175). MR 80m:57031 | Zbl 0418.57014 · Zbl 0418.57014 |

[18] | J. MILNOR , A Unique Decomposition Theorem for 3-Manifolds (Amer. J. Math., 84, 1962 , pp. 1-7). MR 25 #5518 | Zbl 0108.36501 · Zbl 0108.36501 |

[19] | G. MOSTOW , Quasiconformal Mappings in n-Space and the Rigidity of Hyperbolic Space Forms , Publ. Math. Inst. des Hautes Études Scient., Paris, Vol. 34, 1968 , pp. 53-104. Numdam | MR 38 #4679 | Zbl 0189.09402 · Zbl 0189.09402 |

[20] | J. NIELSEN , Untersuchung zur Topologie der Geschlossenen Zweiseitigen , Flachen I, II and III, Acta Mathematica, Vol. 50, 1927 ; Vol. 53, 1929 , and Vol. 58, 1931 . Zbl 0004.27501 | JFM 55.0971.01 · Zbl 0004.27501 |

[21] | J. NIELSEN , Abbildungsklassen endlicher Ordnung (Acta Math., Vol. 75, 1942 , pp. 23-115). MR 7,137a | Zbl 0027.26601 · Zbl 0027.26601 |

[22] | G. PRASAD , Strong rigidity of Q-rank 1 lattices (Invent. Math., Vol. 21, 1973 , pp. 255-286). MR 52 #5875 | Zbl 0264.22009 · Zbl 0264.22009 |

[23] | M. SCHARLEMANN , The Subgroup of \Delta 2 Generated by Automorphisms of Tori (Math. Ann., Vol. 251, 1980 , pp. 263-268). MR 82a:57039 | Zbl 0424.57022 · Zbl 0424.57022 |

[24] | O. TEICHMÜLLER , Extremale quasiconforme Abbildungen und quadratische Differentiale (Abh. Preuss. Akad. Wiss., Vol. 22, 1939 , pp. 187). Zbl 0024.33304 | JFM 66.1252.01 · Zbl 0024.33304 |

[25] | O. TEICHMÜLLER , Bestimmung der extremalen quasiconformem Abbildungen bei geschlossenen orientierten Riemannschen Fläschen , Abh. Preuss. Akad. Wiss, Vol. 4, 1943 , pp. 42. Zbl 0060.23313 · Zbl 0060.23313 |

[26] | W. THURSTON , On the Geometry and Dynamics of Diffeomorphisms of Surfaces I , preprint, Princeton University, 1976 ; for a complete account, see [FLP]. |

[27] | W. THURSTON , The Geometry and Topology of 3-Manifolds (Mimeographed Notes, Princeton University, 1976 - 1979 ). · Zbl 0324.53031 |

[28] | W. THURSTON , Hyperbolic Structures on 3-manifolds , preprint, Princeton University, 1980 . · Zbl 0435.58019 |

[29] | F. WALDHAUSEN , Eine Klasse von 3-dimensionalen Mannigfaltigkeiten, I and II (Inventiones Math., Vol. 3, 1967 , pp. 308-333 and Vol. 4, 1967 , pp. 87-117). MR 38 #3880 | Zbl 0168.44503 · Zbl 0168.44503 |

[30] | F. WALDHAUSEN , On Irreducible Manifolds Which are Sufficently Large (Ann. Math., Vol. 87, 1968 , pp. 56-88). MR 36 #7146 | Zbl 0157.30603 · Zbl 0157.30603 |

[31] | H. WINKELNKEMPER , On equators of Manifolds and the Actions of \theta n (Thesis, Princeton University, 1970 ). |

[32] | H. ZIESCHANG , Uber einfache Kurven auf Vollbrezeln (Abh. Math. Semin. Univ. Hamburg, Vol. 25, 1962 , pp. 231-250). MR 26 #6957 | Zbl 0111.35801 · Zbl 0111.35801 |

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