Bordism invariants of the mapping class group

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin bordism, and we prove that this single representation contains all of the information given by the Johnson homomorphism, the Birman-Craggs homomorphism, and the Morita homomorphism.     

NEW OBSTRUCTIONS FOR THE SURJECTIVITY OF THE JOHNSON HOMOMORPHISM OF THE AUTOMORPHISM GROUP OF A FREE GROUP

In this paper we construct new obstructions for the surjectivityof the Johnson homomorphism of the automorphism group of a freegroup. We also determine the structure of the cokernel of theJohnson homomorphism for degrees 2 and 3.     

An -module structure of the cokernel of the second Johnson homomorphism

The Johnson homomorphisms τ_k (k1) give Abelian quotients of a series of certain subgroups of the mapping class group of a surface. Morita determined the rational image of the second Johnson homomorphism τ₂. In this paper, we study the structure of the torsion part of the cokernel of τ₂. First, we determine the rank of the cokernel over . Although we do it first by computing explicitly, later we improve the proof, using the Birman–Craggs homomorphism, obtained by the classical Rohlin invariant of homology 3-spheres. Since τ₂ is equivariant with respect to the action of the mapping class group, Im τ₂ is -invariant and hence acts on the cokernel. Moreover, computing this action explicitly, we show that the action reduces to that of the finite symplectic group .     

Entries of the group inverse of the Laplacian matrix for generalized Johnson graphs

In this paper, we use graph theoretic properties of generalized Johnson graphs to compute the entries of the group inverse of Laplacian matrices for generalized Johnson graphs. We then use these entries to compute the Zenger function for the group inverse of Laplacian matrices of generalized Johnson graphs.     

On nilpotent ideals in the cohomology ring of a finite group

In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen-Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications.We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero “k-invariant” and study the class related to this obstruction. It turns out that this class is a “universal essential class” for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups.     

Equivalence relations of AF-algebra extensions

In this paper, we consider equivalence relations of C*-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.     

The genus 2 Torelli group is not finitely generated

Let F_g be a closed orientable 2-manifold of genus g. The Torelli group is the kernel of the natural homomorphism from the mapping class group of F₁ to Aut(H₁(F_g)). For g⩾3 the Torelli group has been shown to be finitely generated by Dennis Johnson. We show that it is not finitely generated when g=2.     

Albanese homomorphism of the Chow group of 0-cycles of a real Algebraic variety

We study a certain homomorphism of the Chow group of 0-cycles of degree zero of a real algebraic variety into the group of real points of the Albanese variety; this homomorphism is obtained from the Albanese mapping for the corresponding variety. The kernel of this homomorphism is calculated and estimates for the kernel of the mapping of the torsion groups are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 76–83, January, 1999.     

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.     

On the image of the Burau representation of the IA-automorphism group of a free group

In this paper we study the graded quotients of the lower central series of the image of the IA-automorphism group of a free group by the Burau representation. In particular, we determine their structures for degrees 1 and 2.     

Homology and cohomology groups of the group homomorphism

As is known, the homology and cohomology Massa-Takasu groups for pairs of groups (G, H) are defined by the embedding f: H → G, H < G [2]. In our case, these definitions are extended to an arbitrary group homomorphism φ: Π → G. In particular, we define homology and cohomology groups of the nth order for the homomorphism φ, and if Π = H, we obtain the known theory [2]. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Automorphisms of Green Orders and Their Derived Categories

In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.     

Riemann-Roch for tensor powers

Mapping a locally free module on a scheme to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behavior of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups. Received February 13, 1998; in final form January 12, 1999     

On the strength of Gomory mixed-integer cuts as group cuts

Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra (group cuts) do not seem to be as useful when used in conjunction with GMI cuts. For many practical problem instances, we numerically show that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.     

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

Marked fatgraph complexes and surface automorphisms

Combinatorial aspects of the Torelli–Johnson–Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group K. For K abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target Λ ³ K. Furthermore, the Earle class with coefficients in K is represented by an explicit cocyle.     

On the lower central series of the IA-automorphism group of a free group

In this paper, we study the Johnson homomorphisms ${\tau }_k^{\prime }$ of the automorphism group of a free group of rank $n$, which are defined on the graded quotients of the lower central series of the IA-automorphism group. In particular, we determine the cokernel of ${\tau }_k^{\prime }$ for any $k\ge 2$ and $n\ge k+2$.     

The free topological group over the rationals

In this paper we investigate the topological structure of the Graev free topological group over the rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generated by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some properties of certain canonical maps which are closely related to the topological structure of free groups.