We determine the possible $\mathbb{Z}_{2}$ -cohomology rings of orbit spaces of free actions of $\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold $P(1,n)$ , where $n$ is an odd natural number.

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg–MacLane space.

K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ and in bijection with dominant permutations. He observed that the Schubert cell structure of ${{X}_{\lambda }}$ is indexed by maximal rook placements on the Ferrers board ${{B}_{\lambda \text{ }}}$ , and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$ , ${{B}_{\mu }}$ satisfy the combinatorial condition of rook-equivalence.

We classify the varieties ${{X}_{\lambda }}$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$ . From the results of Carrell and Lieberman there exists a filtration ${{F}_{0}}\subset {{F}_{1}}\subset \cdot \cdot \cdot$ of $A\left( Z \right)$ , the ring of $\mathbb{C}$ -valued functions on $Z$ , such that $\text{Gr }A\left( Z \right)\cong {{H}^{*}}\left( X,\mathbb{C} \right)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$ .

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Let CSn be the flag manifold SO(2n)/U(n). We give a partial classification for the endomorphisms of the cohomology ring H*(CSn; Z) which is very close to a homotopy classification of all selfmaps of CSn. Applications concerning the geometry of the space are discussed.

Let $G$ be a simply-connected, semisimple algebraic group over $k$, an algebraically closed field of characteristic zero. Let ${\cal O}_{\epsilon}[G]$ be the quantized functionalgebra of $G$ at a primitive $\ell$th root of unity$\epsilon$, and let $\overline{{\cal O}_{\epsilon}[G]}$ be the`restricted' quantized function algebra at $\epsilon$,a finite-dimensional $k$-algebra obtained from${\cal O}_{\epsilon}[G]$ by factoring out a centrally generatedideal. It is known that $\overline{{\cal O}_{\epsilon}[G]}$ isa Hopf algebra. We study the cohomology ring $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$,a graded commutative algebra, and, for any finite-dimensional$\overline{{\cal O}_{\epsilon}[G]}$-module $M$, the$\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module$\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$.We prove that for sufficiently large $\ell$ there is an isomorphism of graded algebras\[\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)\cong k[X_1,\ldots ,X_{2N}],\]where each $X_i$ is homogeneous of degree $2$, and $2N$ equals the number of roots associated to $G$. Moreover we show that in this case $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$is a finitely generated $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module.We also show under much less restrictive conditions on$\ell$ that $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$continues to be a finitely generated graded commutative algebra over which $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$ is afinitely generated module.\vspace{6mm}\noindent1991 Mathematics Subject Classification: 16W30, 17B37, 17B56.