Whitehead Group Research Articles

Constraints on heterotic M-theory from s-cobordism

We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homotopy equivalence. In the non-simply connected case, which is important for model building, the interpretation is then in terms of s-cobordism, so that the cobordism is a simple-homotopy equivalence. This gives constraints on the possible cobordisms depending on the fundamental groups and hence provides a characterization of possible compactification manifolds using the Whitehead group – a quotient of algebraic K-theory of the integral group ring of the fundamental group – and a distinguished element, the Whitehead torsion. We also consider the effect on the dynamics via diffeomorphisms and general dimensional reduction, and comment on the effect on F-theory compactifications.

Inductive construction of the p-adic zeta functions for noncommutative p-extensions of exponent p of totally real fields

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet's theory and certain inductive technique. As an application we shall prove a special case of (the p-part of) the non-commutative equivariant Tamagawa number conjecture for critical Tate motives. The main results of this paper give generalisation of those of the preceding paper of the author.

SK1 of graded division algebras

The reduced Whitehead group SK1 of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK1 of a tame valued division algebra over a henselian field coincides with SK1 of its associated graded division algebra. Furthermore, it is shown that SK1 of a graded division algebra is isomorphic to SK1 of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK1 for generic abelian crossed products.

The first almost free Whitehead group

Assume $\mathsf{GCH}$ and that $\kappa$ is the first uncountable cardinal such that there is a non-free $\kappa$-free Abelian Whitehead group of cardinality $\kappa$. We prove that if all $\kappa$-free Abelian groups of cardinality $\kappa$ are Whitehead then $\kappa$ is necessarily an inaccessible cardinal.

Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases

Fix an odd primepp. LetGGbe a compactpp-adic Lie group containing a closed, normal, pro-ppsubgroupHHwhich is abelian and such thatG/HG/His isomorphic to the additive group ofpp-adic integersZp\mathbb {Z}_p. First we assume thatHHis finite and compute the Whitehead group of the Iwasawa algebra,λ(G)\lambda (G), ofGG. We also prove some results about certain localisation ofλ(G)\lambda (G)needed in Iwasawa theory. LetFFbe a totally real number field and letF∞F_{\infty }be an admissiblepp-adic Lie extension ofFFwith Galois groupGG. The computation of the Whitehead groups are used to show that the Main Conjecture for the extensionF∞/FF_{\infty }/Fcan be deduced from certain congruences between abelianpp-adic zeta functions of Deligne and Ribet. We prove these congruences with certain assumptions onGG. This gives a proof of the Main Conjecture in many interesting cases such asZp⋊Zp\mathbb {Z}_p\rtimes \mathbb {Z}_p-extensions.

The Whitehead group and the lower algebraicK–theory of braid groups on 𝕊2and ℝP2

Let M D S or RP 2 . Let PBn.M / and Bn.M / be the pure and the full braid groups of M respectively. If € is any of these groups, we show that € satisfies the Farrell–Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic K–theory of the integral group ring Z€ , for € D PBn.M / . The main results are that for € D PBn.S/ , the Whitehead group of € , z K0.Z€/ and Ki.Z€/ vanish for i 1 and n > 0 . For € D PBn.RP / , the Whitehead group of € vanishes for all n > 0 , z K0.Z€/ vanishes for all n > 0 except for the cases nD 2; 3 and Ki.Z€/ vanishes for all i 1 .

On descent theory and main conjectures in non-commutative Iwasawa theory

AbstractWe develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

Filtration-equivalent [formula omitted]-separable abelian groups of cardinality [formula omitted

We show that it is consistent with ordinary set theory Z F C and the generalized continuum hypothesis that there exist two ℵ 1 -separable abelian groups of cardinality ℵ 1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from Eklof and Mekler (2002) [2].

Actor of a Precrossed Module

Applying the equivalence of the category of precrossed modules with the category of groups with two additional unary operations satisfying the corresponding conditions, the construction of an actor is given in terms of Whitehead group of generalized regular derivations, defined in the article, and the automorphism group of a precrossed module. The analogous approach to this problem in the case of crossed modules leads to the well-known construction given in the works of Lue and Norrie.

The structure of the group G(k[t]): Variations on a theme of Soulé

Following Soule’s ideas [14] we give a presentation of the abstract group G(k[t]) for any semisimple (connected) simply connected absolutely almost simple k–group G. As an application, we give a description of G(k[t]) in terms of direct limits, and show that the Whitehead group and the naive group of connected components of G coincide.

On the Kneser–Tits problem for triality forms

The purpose of this paper is to provide a concrete description of the “Whitehead group” W(k,G) := G(k)/G(k)+ for the simply connected triality forms G of k-rank 1, and to use this description to prove that if k is a global field, then the Kneser–Tits problem for these forms has an affirmative solution.

SK1 of Azumaya algebras over Hensel Pairs

Let A be an Azumaya algebra of constant rank n 2 over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK1(A) coincides with its reduction SK1(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.

A generalization of Whitehead’s problem and its independence

For certain classes of Dedekind domains S we want to characterize S -modules U such that Ext ( U , M ) = 0 for some module S ⊆ M ⊆ Q . We shall call these modules M -Whitehead modules. On the one hand we will show that assuming ( V = L ) all M -Whitehead modules U are S 0 -free, i.e. U ⊗ S 0 is a free S 0 -module where S 0 is the nucleus of M . On the other hand if there is a ladder system on a stationary subset of ω 1 that satisfies 2 -uniformization, then there exists a non- S 0 -free M -Whitehead module. Conversely, we will show that in the special case of Abelian groups the existence of a non- S 0 -free R -Whitehead group–here R is a rational group–implies that there is a ladder system on a stationary subset of ω 1 that satisfies 2 -uniformization.

Algebraic K-theory of groups wreath product with finite groups

The Farrell–Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249–297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249–297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515–526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solvable groups, Proc. London Math. Soc. (3) 87 (2003) 309–336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G ≀ H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515–526; S.K. Roushon, The Farrell–Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups.

ON THE TRIVIALITY OF CERTAIN WHITEHEAD GROUPS

Let Κ be a field of arbitrary characteristic and let g be a prime number different from the characteristic of K. If A is a central simple algebra over Κ whose index is a power of q, we show the triviality of the Whitehead groups SKi(A) and USKi(A) when the cohomological ^-dimension of Κ is at most 2. We give a global version of this result and indicate what can be done in the case where the index of the algebra is a power of the characteristic of the field. Triviality results of the Whitehead group KiSpin(A) are easily derived.

ON THE TRIVIALITY OF CERTAIN WHITEHEAD GROUPS

Let K be a field of arbitrary characteristic and let q be a prime number different from the characteristic of K. If A is a central simple algebra over whose index is a power of q, we show the triviality of the Whitehead groups SK₁ (A) and USK₁(A) when the cohomological q-dimension of is at most 2. We give a global version of this result and indicate what can be done in the case where the index of the algebra is a power of the characteristic of the field. Triviality results of the Whitehead group K₁Spin(A) are easily derived.

Fractional branes in non-compact type IIA orientifolds

We study fractional D-branes in the Type-IIA theory on a non-compact orientifold of the orbifold C^3/Z_3 in the boundary state formalism. We find that the fractional D0-branes of the orbifold theory become unstable due to the presence of a tachyon, while there is a stable D-instanton whose tachyon gets projected out. We propose that the D-instanton is obtained after tachyon condensation. We evidence this by calculating the Whitehead group of the Abelian category of objects corresponding to the boundary states as being isomorphic to Z_2.

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental group of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds.

Reduced K-theory of Azumaya algebras

In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps K i ( F ) → K i ( D ) , where K i are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK 1 . In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

On the Whitehead group of Novikov rings associated to irrational homomorphisms

Given a homomorphism ξ : G → R we show that the natural map i ∗ : Wh ( G ) → Wh ( G ; ξ ) from the Whitehead group of G to the Whitehead group of the Novikov ring is surjective. The group Wh ( G ; ξ ) is of interest for the simple chain homotopy type of the Novikov complex. It also contains the Latour obstruction for the existence of a nonsingular closed 1-form within a fixed cohomology class ξ ∈ H 1 ( M ; R ) , where M is a closed connected smooth manifold.