f : Cn(R ) → U(n)/T n from the configuration space of n ordered distinct points of R to the flag manifold of U(n) which is compatible with the natural action of the symmetric group Σn on both spaces. I also noted in [1] that the action of Σn on the rational cohomology of either space coincides with the regular representation but that the homomorphism f induced by f cannot possibly be an isomorp...

A well-known conjecture about closed hyperbolic 3-manifolds asserts that the first Betti number can be increased without bound by passage to finite sheeted covers. If the manifold is fibred, a strengthening of this conjecture is that the number of fibred faces (see §2.1 for the definition of a fibred face) of the unit ball of the Thurston norm can be made arbitrarily large by passage to finite ...

For a smooth variety X over a base field of characteristic 0 we have its algebraic de Rham cohomology H · dR(X) := H (XZar,Ω · X); for nonsmooth X, one defines H · dR(X) using cohomological descent as in Deligne [D]. If the base field is C, then one has the Betti cohomology H · B(X) := H (Xcl,Q) and a canonical period isomorphism (“integration of algebraic differential forms over topological cy...

For a rooted tree Γ, we consider path ideals of Γ, which are ideals that are generated by all directed paths of a fixed length in Γ. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of Γ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as p...

We consider two varieties associated to a web of quadrics W in P. One is the base locus and the second one is the double cover of P branched along the determinant surface of W . We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W co...

In this paper we construct explicit examples of both closed and non-compact nite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group (also known as the cut number) is bounded below by one-third the rst Betti number. AMS Classi cation 57M05; 57M50, 20F34

Let NA be the monoid generated byA = {a1, . . . ,an} ⊆ Z.We introduce the homogeneous catenary degree of NA as the smallest N ∈ N with the following property: for each a ∈ NA and any two factorizations u,v of a, there exists factorizations u = w1, . . . ,wt = v of a such that, for every k, d(wk,wk+1) ≤ N, where d is the usual distance between factorizations, and the length of wk, |wk|, is less ...

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K̂, elements of which, modulo a complex torus of dimension the sum of the odd Betti n...

If M is a compact 3-manifold whose first betti number is 1, and N is a compact 3-manifold such that π1N and π1M have the same finite quotients, then M fibres over the circle if and only if N does. We prove that groups of the form F2 o Z are distinguished from one another by their profinite completions. Thus, regardless of betti number, if M and N are punctured torus bundles over the circle and ...

This paper investigates "what's in a number", i.e. how natural numbers are pronounced in several European languages. As regards reading numbers above 20, 29 languages read the decade first and then the digit, e.g. twenty-four, and 10 languages read the digit before the decade, e.g. four-and-twenty. Two languages, Norwegian and Czech, use both systems, and 9 languages use (partly) a vigesimal sy...