Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

A New Proof of a Theorem of Suslin

In this paper we compute the group H₂(SL₂(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H₂(SL₂, (F)) is the fiber product of λ_*:K₂(F)→ I²(F)/I³(F) and σ: I²(F) → I²(F)/I³(F) where λ_* and σ map onto I²(F)/I³(F). (Received: February 2003)     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields K=k(D)over k=Fq(T)．For five series of real quadratic function fields K,the bounds of h(D)are given more explicitly,e．g.,if D=F^2 C．then h(D)≥degF／degP;if D=(SG)^2 cS．then h(D)≥degS／degP;if D=(A^m a)^2 A,then h(D)≥degA／degP,where P is an irreducible polynomial splitting in K,c∈Fq．In addition,three types of quadratic function fields K are found to have ideal class numbers bigger than one．     

On the ring of invariants ofF_{2^n }^*

In [1] the first and last authors studied a decomposition ofH ^*(R P ^∞×…×R P ^∞;F ₂) into modules over the Steenrod algebra obtained from an action of the cyclic group . Here a minimal set of generators for the ring of invariants is characterized and counted by analyzing the associated ring of Laurent polynomials. A structure theorem for the ring of invariant Laurent polynomials is given and a ‘destabilisation cancels localisation’ theorem is obtained. The authors gratefully acknowledge the support of NSERC. 1980 Mathematics Subject classification, 13F20, 55. Keywords: Invariant theory, Steenrod algebra.     

Proof of a tiling conjecture of Komlós

A conjecture of Komlós states that for every graph H, there is a constant K such that if G is any n‐vertex graph of minimum degree at least (1 ? (1/χ_cr(H)))n, where χ_cr(H) denotes the critical chromatic number of H, then G contains an H‐matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 180–205, 2003     

On the restriction of cross characteristic representations of 2 F 4(q) to proper subgroups

We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that ${K \in \{^2F_4(2), ^2F_4(2)'\} }We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that K ? {²F₄(2), ²F₄(2)￠}{K \in \{^2F_4(2), ^2F_4(2)'\} } , H is a proper subgroup of K, and V is a representation of K in odd characteristic restricting absolutely irreducibly to H.     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.     

Field invariants under totally positive quadratic extensions

K = F(√d) is a formally real field and a totally positive quadratic extension of F. A decomposition theorem for quadratic forms in Fed (K) is given. The invariants r(q) and ud(KF) are defined and relations between the invariants β_F(i), β_K(i), ud(F), ud(K), l(F), l(K) are studied, using the theory of quadratic forms.     

Minimal underlying division rings of sets of points of a projective space

Let V be a vector space over a division ring K. Let P be a spanning set of points in Σ:=PG(V). Denote by K(P) the family of sub-division rings F of K having the property that there exists a basis B_F of V such that all points of P are represented as F-linear combinations of B_F. We prove that when K is commutative, then K(P) admits a least element. When K is not commutative, then, in general, K(P) does not admit a minimal element. However we prove that under certain very mild conditions on P, any two minimal elements of K(P) are conjugate in K, and if K is a quaternion division algebra then K(P) admits a minimal element.     

Complexes of graph homomorphisms

Hom(G, H) is a polyhedral complex defined for any two undirected graphsG andH. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom(K _m, K_n) is homotopy equivalent to a wedge of (n−m)-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graphG, and integersm≥2 andk≥−1, we have ϖ ₁ ^k (Hom(K _m, G))≠0, thenχ(G)≥k+m; here ℤ₂-action is induced by the swapping of two vertices inK _m, and ϖ₁ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom(G, H) induces a homotopy equivalence. It then follows that Hom(F, K _n) is homotopy equivalent to a direct product of (n−2)-dimensional spheres, while Hom(F, K _n) is homotopy equivalent to a wedge of spheres, whereF is an arbitrary forest andF is its complement. The second author acknowledges support by the University of Washington, Seattle, the Swiss National Science Foundation Grant PP002-102738/1, the University of Bern, and the Royal Institute of Technology, Stockholm.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

On counting twists of a character appearing in its associated Weil representation

Consider an irreducible, admissible representation π of GL(2,F) whose restriction to GL(2,F)⁺ breaks up as a sum of two irreducible representations π ₊ + π −. If π = r _θ , the Weil representation of GL(2,F) attached to a character θ of K ^* does not factor through the norm map from K to F, then c ? [^(K^*)]\chi\in \widehat{K^*} with (c. q^-1)| _{F ^*} =w _K/F(\chi . \theta ^{-1})\vert _{ F^{ * }}=\omega _{ {K/F}} occurs in r _{θ  +} if and only if e(qc^-1,y₀)=e([`(q)]c^-1,y₀)=1\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\overline \theta\chi^{-1},\psi_0)=1 and in r _θ− if and only if both the epsilon factors are − 1. But given a conductor n, can we say precisely how many such χ will appear in π? We calculate the number of such characters at each given conductor n in this work.     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

Finiteness of integral values for the ratio of two linear recurrences

Let {F(n)} _{n ∈ N} ,{G(n)} _{n ∈ N} , be linear recurrent sequences. In this paper we are concerned with the well-known diophantine problem of the finiteness of the set ? of natural numbers n such that F(n)/G(n) is an integer. In this direction we have for instance a deep theorem of van der Poorten; solving a conjecture of Pisot, he established that if ? coincides with N, then {F(n)/G(n)} _{n ∈ N} is itself a linear recurrence sequence. Here we shall prove that if ? is an infinite set, then there exists a nonzero polynomial P such that P(n)F(n)/G(n) coincides with a linear recurrence for all n in a suitable arithmetic progression. Examples like F(n)=2 ⁿ -2, G(n)=n+2 ⁿ +(-2) ⁿ , show that our conclusion is in a sense best-possible. In the proofs we introduce a new method to cope with a notorious crucial difficulty related to the existence of a so-called dominant root. In an appendix we shall also prove a zero-density result for ? in the cases when the polynomial P cannot be taken a constant. Oblatum 12-XI-2001 & 31-I-2002?Published online: 29 April 2002     

On some infinite dimensional linear groups

Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.      

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2 ^X generated by F, let G be a Hausdorff commutative topological lattice group and let rba_F(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rba_F(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.     

Existence and comparison results for nonlinear volterra integral equations in a banach space

Let L ⁰(ΩA,P) be the space of equivalent classes of random variables defined on a probability space (Ω,A,P). Let H be the closed subspace of L ⁰(Ω,A,P) spanned by a sequence of i.i.d. (independent and identically distributed) random variables having the symmetric nondegenerate law F. It is proved that H is linearly homeomorphic to l _p for 0<p≤2 if F belongs to the domain of normal attraction of symmetric stable law withexponent p.