Dold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${\prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.

During the last decades, the structure of mod-2 cohomology of the Steenrod ring $\mathscr {A}$ became a major subject in Algebraic topology. One of the most direct attempt in studying this cohomology by means of modular representations of the general linear groups was the surprising work [Math. Z. 202 (1989), 493–523] by William Singer, which introduced a homomorphism, the so-called algebraic transfer, mapping from the coinvariants of certain representation of the general linear group to mod-2 cohomology group of the ring $\mathscr A.$ He conjectured that this transfer is a monomorphism. In this work, we prove Singer's conjecture for homological degree $4.$

The algebraic EHP sequences, algebraic analogues of the EHP sequences in homotopy theory, are important tools in algebraic topology. This note will outline two new proofs of the existence of the algebraic EHP sequences. The first proof is derived from the minimal injective resolution of the reduced singular cohomology of spheres, and the second one follows Bousfield's idea using the loop functor of unstable modules.

For a prime number $p$ , we show that differentials $d_{n}$ in the motivic cohomology spectral sequence with $p$ -local coefficients vanish unless $p-1$ divides $n-1$ . We obtain an explicit formula for the first non-trivial differential $d_{p}$ , expressing it in terms of motivic Steenrod $p$ -power operations and Bockstein maps. To this end, we compute the algebra of operations of weight $p-1$ with $p$ -local coefficients. Finally, we construct examples of varieties having non-trivial differentials $d_{p}$ in their motivic cohomology spectral sequences.

We consider classifying spaces of a family of p-groups and prove that mod p cohomology enriched with Bockstein spectral sequences determines their homotopy type among p-completed CW-complexes.

We make some computations in stable motivic homotopy theory over Spec ℂ, completed at 2. Using homotopy fixed points and the algebraic K-theory spectrum, we construct over ℂ a motivic analogue of the real K-theory spectrum KO. We also establish a theory of motivic connective covers over ℂ to obtain a motivic version of ko. We establish an Adams spectral sequence for computing motivic ko-homology. The E2-term of this spectral sequence involves Ext groups over the subalgebra A(1) of the motivic Steenrod algebra. We make several explicit computations of these E2-terms in interesting special cases.

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

Let $p$ be an odd prime. The primary purpose of this paper is to determine the excess of the conjugates of the Steenrod operations $\mrm{P}[k;f]$, which are defined as $\mrm{P}[k;f]:=\mrm{P}(p^{k-1}f)\cdot\mrm{P}(p^{k-2}f)\cdot\cdots\cdot\mrm{P}(pf)\cdot\mrm{P}(f)$. The result is then used to obtain sufficient conditions for an element in the polynomial algebra $\mathbb{F}_p[x_1,\dots,x_s]$ to be in the image under the standard action of the Steenrod algebra. Results and methods are generalizations of previous work by Judith Silverman and by myself with Judith Silverman.

AMS 2000 Mathematics subject classification: Primary 55S10. Secondary 55S05

The Novikov-Landweber algebra and the Steenrod algebra are set up in terms of the primitive differential operators $D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way on the integral polynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. A commutative wedge product $\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\vee r}$ is divisible by $r!$ as an integral operator. The divided differential operator algebra $\cal D$ is generated over the integers by the divided operators $\frac{D_k^{\vee r}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic to the abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect to the composition and wedge products. Under composition $\cal D$ is isomorphic to the Landweber-Novikov algebra. A Hopf sub-algebra is generated under composition by the integral Steenrod squares $SQ^r = \frac{D_1^{\vee r}}{r!}$ and reduces mod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.