The Quarterly Journal of Mathematics - current issue

HOCHSCHILD HOMOLOGY AND COHOMOLOGY OF {ell}1(ZFormula)

Choi, Y. — Mon, 08 Feb 2010 07:48:30 PST

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ¹(Z^k₊) [F. Gourdeau, Z. A. Lykova and M. C. White, A Künneth formula in topological homology and its applications to the simplicial cohomology of ¹(Z^k₊), Studia Math. 166 (2005), 29–54], we investigate what can be said for the cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of the Harrison homology and cohomology in the context of Banach algebras and a development of some of its basic features. As an application of our techniques, we reprove some known results on second-degree cohomology.

ON SUMS OF 13 'ALMOST EQUAL' CUBES

Daemen, D. — Mon, 08 Feb 2010 07:48:30 PST

A simple proof of a special case is presented in Waring's problem on sums of 13 cubes localized close to their average size, which currently seems to be out of reach for the circle method.

SURFACES WITH CONSTANT MEAN CURVATURE IN RIEMANNIAN PRODUCTS

De Lira, J. H. S., Vitorio, F. A. — Mon, 08 Feb 2010 07:48:30 PST

We prove the existence of holomorphic quadratic differentials for surfaces with parallel mean curvature in some four-dimensional products of space forms. These differentials are then used to characterize spheres with parallel mean curvature immersed into these spaces.

THE EULER CLASS OF A SUBSET COMPLEX

Guclukan, A., Yalcin, E. — Mon, 08 Feb 2010 07:48:30 PST

The subset complex (G) of a finite group G is defined as the simplicial complex whose simplices are non-empty subsets of G. The oriented chain complex of (G) gives a ZG-module extension of Z by tilde;, where tilde; is a copy of integers on which G acts via the sign representation of the regular representation. The extension class _G Ext_ZG^|G|–1 (Z, tilde;) of this extension is called the Ext class or the Euler class of the subset complex (G). This class was first introduced by Reiner and Webb [The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291–327] who also raised the following question: What are the finite groups for which _G is non-zero?

In this paper, we answer this question completely. We show that _G is non-zero if and only if G is an elementary abelian p-group or G is isomorphic to Z/9, Z/4 x Z/4 or (Z/2)ⁿ x Z/4 for some integer n ≥ 0. We obtain this result by first showing that _G is zero when G is a non-abelian group, then by calculating _G for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex (G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.

We also give some applications of our results to group cohomology, to filtrations of modules and to the existence of Borsuk–Ulam type theorems.

KIRWAN SURJECTIVITY IN K-THEORY FOR HAMILTONIAN LOOP GROUP QUOTIENTS

Harada, M., Selick, P. — Mon, 08 Feb 2010 07:48:30 PST

Let G be a compact Lie group and LG be its associated loop group. The main result of this article is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology by R. Bott, S. Tolman and J. Weitsman, Surjectivity for Hamiltonian loop group spaces, Invent. Math. 155 (2004), 225–251, math.DG/0210036. Our proof techniques differ from that of Bott et al. in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step towards a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.

ON SOME CONFORMAL MINIMAL 2-SPHERES IN A COMPLEX PROJECTIVE SPACE

Jiao, X., Peng, J. — Mon, 08 Feb 2010 07:48:30 PST

In this paper, the geometry of a linearly full conformal minimal 2-sphere S² immersed in a complex projective space CPⁿ which satisfies various conditions is studied. Let ₁(p) be the first normal space of S² at the point p, and let T_p S² = ₁(p) ₂(p) for p S². We prove that S² is of constant Kähler angle if and only if J₁(p) T_p S² for all p S², where J is the complex structure of CPⁿ. Furthermore, we prove that (i) S² is totally geodesic in CP² if J ₁(p) T_p S² for all p S²; (ii) S² is either a holomorphic curve in CPⁿ or the first element of the Veronese sequence, up to an isometry of CPⁿ, if J₁(p) ₁(p) for all p S²; (iii) S² is totally real if J₁(p) ₂(p) for all p S². It is also proved that S² is either an element of the Veronese sequence in CP² or a totally real curve of constant curvature 1/3 in CP⁴ if its second fundamental form is parallel.

A NOTE ON BELYI'S THEOREM FOR KLEIN SURFACES

Kock, B., Lau, E. — Mon, 08 Feb 2010 07:48:30 PST

Singerman and the first named author have recently developed a real Belyi theory, leaving open a particular case in the proof of Belyi's theorem for Klein surfaces. We answer their question affirmatively by a descent argument which turns out to extend to a much more general context.

A NOTE ON THE SUM OF THE FIRST n PRIMES

Matomaki, K. — Mon, 08 Feb 2010 07:48:30 PST

We show that the arithmetic mean of the first n primes is an integer for << N^19/24+ numbers n ≤ N. This follows from showing that the discrepancy of the sequence consisting of the arithmetic means is << N^–5/24+.

DISTRIBUTION OF ANGLES IN HYPERBOLIC LATTICES

Risager, M. S., Truelsen, J. L. — Mon, 08 Feb 2010 07:48:30 PST

We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result from the study by Boca.

ABSOLUTE CONTINUITY ON C*-ALGEBRAS

Saito, K., Wright, J. D. M. — Mon, 08 Feb 2010 07:48:30 PST

In an earlier work the notion of absolute continuity was extended from finitely additive measures to non-commutative C*-algebras. But to obtain a generalisation of the Vitali–Hahn–Saks theorem valid for all C*-algebras it was necessary to introduce ‘weak’ and ‘strong’ absolute continuity. For commutative algebras, these two notions of absolute continuity coincide but, given recent work by Chetcuti and Hamhalter, it is reasonable to ask if there are wider classes of C*-algebras for which weak and strong absolute continuity coincide.We show here that this is not true. If weak and strong absolute continuity coincide for a given algebra then the algebra must be commutative.