Anthony Bak

Normality for elementary subgroup functors

We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E . We show under a general condition which is easily checked in practice that the elementary subgroup E ( M ) of G ( M ) is normal for all quasi-weak Noetherian objects M in the source category of G . This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

Stability for Hermitian K1

The general Hermitian group GH2n and its elementary subgroup EH2n are the analogs in the theory of Hermitian forms of the general linear group GLn and its elementary subgroup En. This article proves that the canonical map GH2n/EH2n→GH2(n+1)/EH2(n+1) is an isomorphism whenever n is large with respect to a suitable stable range condition for rings with involution.

Local-global principle for transvection groups

In this article we extend the validity of Suslins Local-Global Principle for the elementary transvection subgroup of the general linear group GL n (R), the symplectic group Sp 2n (R), and the orthogonal group O 2n (R), where n > 2, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut(P) of either a projective module P of global rank > 0 and constant local rank > 2, or of a nonsingular symplectic or orthogonal module P of global hyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslins results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank > 0 is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET(P) is normal in Aut(P), that ET(P) = T(P), where the latter denotes the full transvection subgroup of Aut(P), and that the unstable K 1 -group K 1 (Aut(P)) = Aut(P)/ET(P) = Aut(P)/T(P) is nilpotent by abelian, provided R has finite stable dimension. The last result extends previous ones of Bak and Hazrat for GL n (R), Sp 2n (R), and O 2n (R). An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C 1 -fields. We refer the reader to that article for more details.

EQUIVARIANT SURGERY WITH MIDDLE-DIMENSIONAL SINGULAR SETS. I

Let G be a nite group. Let f : X ! Y be a k-connected, degree 1, G-framed map of simply connected, closed, oriented, smooth manifolds X and Y of dimension 2k = 6. Under the assumption that the dimension of the singular set of the action of G on X is at most k, we construct an abelian group W (G; Y ) and an element (f) 2 W (G; Y ), called the surgery obstruction of f such that the vanishing of (f) in W (G; Y ) guarantees that f can converted by G-surgery to a homotopy equivalence. Acknowledgement. We wish to express our gratitude to I. Madsen for stressing the impor- tance of the problem above. We are also grateful to E. Laitinen and K. Pawa lowski for their suggestions concerning brushing up the manuscript. The second author gratefully acknowl- edges support by the Japan Association for Mathematical Sciences and a Grant-in-Aid for Scientic Research from the Ministry of Education, Science and Culture, Japan.

SPLITTING ALONG SUBMANIFOLDS AND L-SPECTRA

AbstractThe problem of splitting of homotopy equivalence along a submanifold is closely related to surgeries of submanifolds and exact sequences in surgery theory. We describe possibilities and methods of application of

The computation of surgery groups of odd torsion groups

\mathbb{L}

The dimension of spheres with smooth one fixed point actions

-spectra for the investigation of the problem of splitting of (simple) homotopy equivalence of manifolds along submanifolds. This approach naturally leads us to commutative diagrams of exact sequences, which play an important role in calculational problems of surgery theory.

Current trends in transformation groups

(1978). The computation of even dimension surgery groups of odd torsion groups. Communications in Algebra: Vol. 6, No. 14, pp. 1393-1458.