The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost Hopf rings

We describe a Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( B_n; \mathbb{Z}_2 \right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( D_n; \mathbb{Z}_2 \right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements $\mathbb{Z}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.Comment: 32 page

Stable characteristic classes of smooth manifold bundles

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.Comment: 52 page

A summary of research in elementary school social studies (1948-1950)

Thesis (Ed.M.)--Boston Universit

Orbifold Quantum Cohomology

This is a research announcement of the theory of orbifold quantum cohomology.Comment: Revised version, adding more reference