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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independently K. Johannson and D. Johnson have shown that these methods are insufficient to determine $$\Delta_{2+}$$ (unpublished).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 $$\Delta^ p\!_ 2\to \Delta_ 2$$.
Reviewer: N.Ivanov

##### 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
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