Twisted homology of symmetric groups

We study the homology of symmetric groups with coefficients coming from the functor . We are primarily interested in the limit where . Our main goal is to compare the described above situation with the case of general linear groups.

     

Twisted Novikov homology of complex hypersurface complements

We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self‐contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov homology groups associated to positive cohomology classes of degree one defined on the complement.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

Einstein solvmanifolds with free nilradical

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the Abelian and the two-step ones, there are only six others: is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradical obtained in the paper.      

On the mirabolic Lie algebra \mathfrak{p}_n

We consider the Lie algebra \(\mathfrak{g} = \mathfrak{p}_n \) of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P _n .     

On the homology spectral sequence for topological Hochschild homology

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of -algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower' notation for the Dyer-Lashof algebra.

     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Classification of solvable Leibniz algebras with null-filiform nilradical

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.     

Intersection Theory for Twisted Cycles

In this paper we generalize two Noether-Lefschetz theorems for surfaces by L. Ein to varieties of arbitrary even dimension. Our results also generalize the classical Noether-Lefschetz theorem for complete intersections of even dimension in ?_n.     

Twisted index theory for foliations

For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum–Connes assembly maps for foliations with a twisting on the space of leaves.     

Twisted Automorphisms and Twisted Right Gyrogroups

In this paper we make an attempt to study right loops (S, o) in which, for each y ∈ S, the map σ _y from the inner mapping group G _S of (S, o) to itself given by σ _y (h)(x)o h(y) = h(xoy), x ∈ S, h ∈ G _S is a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups.     

Contact and Frobenius solvable Lie algebras with abelian nilradical

The aim of this work is to characterize the families of Frobenius (respectively, contact) solvable Lie algebras that satisfies the following condition: 𝔤 = 𝔥?V, where 𝔥?𝔤𝔩(V), |dim V?dim 𝔤|≤1 and NilRad(𝔤) = V, V being a finite dimensional vector space. In particular, it is proved that every complex Frobenius solvable Lie algebra is decomposable, whereas that in the real case there are only two indecomposable Frobenius solvable Lie algebras.     

Intersection Numbers for Twisted Homology

We prove by combinatorial means new formulae for intersection numbers of twisted homology of hyperplane complements, thereby solving a conjecture of Aomoto.     

Complete Reducibility for the Twisted Affine Lie Algebras

We prove complete reducibility theorem for integrable modules for the twisted affine Lie algebras where the central element acts non-trivially.     

Twisted Conjugacy Classes for Polyfree Groups

Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.     

Twisted Planes

Let k be a commutative ring. We find and characterize a new family of twisted planes (i.e., associative unitary k-algebra structures on the k-module k[X, Y], having k[X] and k[Y] as subalgebras). Similar results are obtained for the k-module of two variables power series k[[X, Y]].