### Article

## On non-immersibility of RP10 to R15

In 1965, P.F. Baum and W. Browder proved that RP10 cannot be immersed to R15. Going alternative way, we investigate this problem using U. Koschorke’ singularity approach. In this paper, we simplify and analyze the corresponding obstruction group.

We prove the following: (1) the existence, for every integer *n* ≥ 4, of a noncompact
smooth *n*-dimensional topological manifold whose diffeomorphism group contains an isomorphic
copy of every finitely presented group; (2) a finiteness theorem for finite simple subgroups of
diffeomorphism groups of compact smooth topological manifolds.

We discuss numerical schemes of finite element method for solving the continuum mechanics problems. Previously a method of acceleration of calculations was developed which uses the simplicial mesh inscribed in the original cubic cell partition of a three-dimensional body. In this paper we show that the obstacle to the construction of this design may be described in terms of homology groups modulo 2. The main goal of the paper is to develop a method of removing this obstacle. The reaching of the goal is based on efficient algorithms for computing bases of the homology groups which are dual with respect to the intersection form.

We prove the following: (1) the existence, for every integer *n* ≥ 4, of a noncompact
smooth *n*-dimensional topological manifold whose diffeomorphism group contains an isomorphic
copy of every finitely presented group; (2) a finiteness theorem for finite simple subgroups of
diffeomorphism groups of compact smooth topological manifolds.

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.