Volume 2012, Issue 669

Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let ℒ BG  ≔ map( S 1 ,  BG ) be the free loop space of BG , i.e. the space of continuous maps from the circle S 1 to BG . The purpose of this paper is to study the singular homology H * (ℒ BG ) of this loop space. We prove that when taken with coefficients in a field the homology of ℒ BG is a homological conformal field theory. As a byproduct of our Main Theorem, we get a Batalin–Vilkovisky algebra structure on the cohomology H * (ℒ BG ). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH * ( S * ( G ),  S * ( G )) of the singular chains of G is a Batalin–Vilkovisky algebra.

Let X be the quotient of an irreducible bounded symmetric domain Ω by a lattice. In order to characterize algebraic correspondences on X commuting with exterior Hecke correspondences, Clozel–Ullmo studied certain germs of measure-preserving maps from (Ω; 0) into its Cartesian products, proving that such maps are totally geodesic when dim( X ) = 1. Here we prove total geodesy when dim(Ω) ≧ 2 by methods of analytic continuation. For B n , n  ≧ 2, total geodesy follows then from Alexander's theorem. When rank(Ω) ≧ 2, we deduce total geodesy from Alexander-type theorems, especially from a new Alexander-type theorem involving Reg(∂Ω) in place of the Shilov boundary.

Consider a domain Ω ⊂ ℝ n with possibly non compact but uniform C 3 -boundary and assume that the Helmholtz projection P exists on L p (Ω) for some 1 <  p  < ∞. It is shown that the Stokes operator in L p (Ω) generates an analytic semigroup on admitting maximal L q - L p -regularity. Moreover, for there exists a unique local mild solution to the Navier–Stokes equations on domains of this form provided p  >  n .

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have “too many” arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3-manifold groups in all but a few exceptional cases.

In this paper, we give a Thomae type formula for K 3 surfaces X given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type I 22 and period integrals of X . Moreover, we express the period integrals by using the hypergeometric function F S of four variables. As applications of our main theorem, we define ℝ 4 -valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function F S .

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot K is n -adjacent to a fibered knot K ′, for some n  > 1, then either the genus of K is larger than that of K ′ or K is isotopic to K ′.

The construction of the Leavitt path algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E . The new algebras, L K ( E ,  C ), are analyzed in terms of their homology, ideal theory, and K -theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid 𝒱( L K ( E ,  C )) of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of L K ( E ,  C ) is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of E 0 and a subset of C . Necessary conditions for 𝒱( L K ( E ,  C )) to be a refinement monoid are developed, together with a construction that embeds ( E ,  C ) in a separated graph ( E + ,  C + ) such that 𝒱( L K ( E + ,  C + )) has refinement.

We prove Brauer's height zero conjecture on blocks of finite groups for every 2-block of maximal defect.