### Article

## Topological classification of Morse-Smale diffeomorphisms without heteroclinic intersection

We study the class G(Mn)of orientation-preserving Morse–Smale diffeomorfisms on a connected closed smooth manifold Mn of dimension n>3 which is defined by the following condition: for any f∈G(Mn)the invariant manifolds of saddle periodic points have dimension1and(n−1)and contain no heteroclinic intersections. For diffeomorfisms in G(Mn) we establish the topoligical type of the supporting manifold which is determined by the relation between the numbers of saddle and node periodic orbits and obtain necessary and sufficient conditions for topological conjugacy.

It is shown that if a closed smooth orientable manifold *M**n*, *n* ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.