### 3-dimensional cohomology of the mod p Steenrod algebra.

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It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family ${w}_{\alpha}^{H}\left(X\right)$ of elements of the groups ${K}_{0}\left(\mathbb{Z}\left[{\pi}_{0}{\left(WH\left(X\right)\right)}_{\alpha}^{*}\right]\right)$. We prove that every family ${w}_{\alpha}^{H}$ of elements of the groups ${K}_{0}\left(\mathbb{Z}\left[{\pi}_{0}{\left(WH\left(X\right)\right)}_{\alpha}^{*}\right]\right)$ can be realized as the family of equivariant finiteness obstructions ${w}_{\alpha}^{H}\left(X\right)$ of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s obstruction...

We will give a condition characterizing spaces X with SNT(X) = {[X]} which generalizes the corresponding result of McGibbon and Moller [8] for rational H-spaces.

The paper studies applications of ${C}^{*}$-algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $AF$-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $AF$-algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$, $3$ and $4$. In conclusion, we consider two numerical examples illustrating our main results.

We define an isotopy invariant of embeddings $N\to {\mathbb{R}}^{m}$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...

We show that the property of having only vanishing triple Massey products in equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular it can be viewed as a partial answer to a question posed by Allday and Puppe about finding conditions ensuring the "formality" of G-actions.

A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré-Birkhoff-Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime.