121,556 research outputs found
Third homology of general linear groups
The third homology group of GL_n(R) is studied, where R is a `ring with many
units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq
K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then
H_3(GL_2(R), Q) --> H_3(GL_3(R), Q) is injective. If R is commutative, Q can be
replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an
infinite field R such that R*=R*^2), we get a better result that H_3(GL_2(R),
Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is
injective. As an application we study the third homology group of SL_2(R) and
the indecomposable part of K_3(R).Comment: 24 pages, Latex, new title, some results are generalized, added
reference
Euler class groups, and the homology of elementary and special linear groups
We prove homology stability for elementary and special linear groups over
rings with many units improving known stability ranges. Our result implies
stability for unstable Quillen K-groups and proves a conjecture of Bass. For
commutative local rings with infinite residue fields, we show that the
obstruction to further stability is given by Milnor-Witt K-theory. As an
application we construct Euler classes of projective modules with values in the
cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative
noetherian rings with infinite residue fields we show that the vanishing of the
Euler class is necessary and sufficient for a projective module P of rank d to
split off a rank 1 free direct summand. Along the way we obtain a new
presentation of Milnor-Witt K-theory.Comment: 64 pages. Revised Section 5. Comments welcome
The rational stable homology of mapping class groups of universal nil-manifolds
We compute the rational stable homology of the automorphism groups of free
nilpotent groups. These groups interpolate between the general linear groups
over the ring of integers and the automorphism groups of free groups, and we
employ functor homology to reduce to the abelian case. As an application, we
also compute the rational stable homology of the outer automorphism groups and
of the mapping class groups of the associated aspherical nil-manifolds in the
TOP, PL, and DIFF categories.Comment: 25 pages, will appear at Annales de l'Institut Fourie
Kostant homology formulas for oscillator modules of Lie superalgebras
We provide a systematic approach to obtain formulas for characters and
Kostant -homology groups of the oscillator modules of the finite
dimensional general linear and ortho-symplectic superalgebras, via Howe
dualities for infinite dimensional Lie algebras. Specializing these Lie
superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant
homology groups of unitarizable highest weight representations of Hermitian
symmetric pairs. In addition, two new reductive dual pairs related to the
above-mentioned -homology computation are worked out
On finite simple and nonsolvable groups acting on homology 4-spheres
The only finite nonabelian simple group acting on a homology 3-sphere -
necessarily non-freely - is the dodecahedral group (in analogy, the only finite perfect group acting freely on a
homology 3-sphere is the binary dodecahedral group ). In the present paper we show that the only finite simple groups
acting on a homology 4-sphere, and in particular on the 4-sphere, are the
alternating or linear fractional groups groups
and . From this we deduce a short list of groups
which contains all finite nonsolvable groups admitting an action on a homology
4-spheres.Comment: 15 page
The third homology of the special linear group of a field
We prove that for any infinite field homology stability for the third
integral homology of the special linear groups begins at . When
the cokernel of the map from the third homology of to the third
homology of is naturally isomorphic to the square of Milnor . We
discuss applications to the indecomposable of the field and to
Milnor-Witt K-theory.Comment: PDFLatex, 21 page
- …