We study the natural map η between a group of binary planar trees whose leaves are labeled by elements of a free abelian group H and a certain group D(H) derived from the free Lie algebra over H. Both of these groups arise in several different topological contexts. η is known to be an isomorphism over Q, but not over Z. We determine its cokernel and attack the conjecture that it is injective.

Let $F$ be a field of characteristic 2 and let $X$ smooth projective quadric dimension $\ge 1$ over $F$. We study the unramified cohomology groups with 2-primary torsion coefficients in degrees 3. determine completely kernel cokernel natural map from to $X$. This extends results different obtained by Kahn, Rost Sujatha nineteen-nineties.

We prove that for any infinite field F , the map H3(SLn(F ), Z)→ H3(SLn+1(F ), Z) is an isomorphism for all n ≥ 3. When n = 2 the cokernel of this map is naturally isomorphic to 2 · K 3 (F ), where K n (F ) is the nth Milnor K-group of F . We deduce that the natural homomorphism from H3(SL2(F ), Z) to the indecomposable K3 of F , K3(F ), is surjective for any infinite field F .

We study the cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto-Satoh and Conant-Kassabov-Vogtmann is given, and using technology from the author’s work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes se...

Dedicated to Yuri Manin on the occasion of his 70th birthday O simili o dissimili che sieno questi mondi non con minor raggione sarebe bene a l'uno l'essere che a l'altro Giordano Bruno – De l'infinito, universo e mondi Contents 1. Introduction 2 2. A look at the Weil proof 4 2.1. Correspondences and divisors 7 2.2. The explicit formula 8 2.3. Riemann–Roch and positivity 9 2.4. A tentative dict...

Dedicated to Yuri Manin on the occasion of his 70th birthday O simili o dissimili che sieno questi mondi non con minor raggione sarebe bene a l'uno l'essere che a l'altro Giordano Bruno – De l'infinito, universo e mondi Contents 1. Introduction 2 2. A look at the Weil proof 4 2.1. Correspondences and divisors 6 2.2. The explicit formula 8 2.3. Riemann–Roch and positivity 9 2.4. A tentative dict...

Let G be the group scheme SL d + 1 over Z and let Q parabolic subgroup corresponding to simple roots ? 2 , ? ? . Then / is -scheme of partial flags { D ? H V } We will calculate cohomology modules line bundles this flag scheme. prove that only non-trivial ones are isomorphic kernel or cokernel certain matrices with multinomial coefficients.