I t has been long known that the cohomology H (A) of the mod 2 Steenrod algebra A admits squaring operations. (For example, see [ó].) Since H(A) is isomorphic to the £2 term of the mod 2 Adams spectral sequence [2], it is natural to inquire as to the relation of these squaring operations to the structure of the Adams spectral sequence. In this note we announce some results of this type extendin...

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U(1, n−1). These cohomology theories of topological automorphic forms (TAF ) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We c...

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

We propose a simple method for constructing formal deformations of differential graded algebras in the category minimal $A_\infty$-algebras. The basis our approach is provided by Gerstenhaber algebra structure on $A_\infty$-cohomology, which we define terms brace operations. As an example, construct $A_\infty$-algebra from Weyl-Moyal $\ast$-product polynomial functions.

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$  are  finitely generated  $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)...

Operations on the cohomology of spaces are important tools enhancing descriptive power this computable invariant. For with mod 2 coefficients, Steenrod squares most significant these operations. Their effective computation relies formulas defining a cup-$i$ construction, structure (co)chains which is in its own right, having connections to lattice field theory, convex geometry and higher catego...

We prove a geometrical version of Herbert’s theorem [4] by considering the self-intersection immersions of a self-transverse immersion up to bordism. This generalises Herbert’s theorem to additional cohomology theories and gives a commutative diagram in the homotopy of Thom complexes. The proof is based on Herbert’s and uses Koschorke and Sanderson’s operations [6] and the fact that bordism of ...

This paper is devoted to qualgebras and squandles, which are quandles enriched with a compatible binary/unary operation. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. Topologically, qualgebras emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an “algebraization” of knots; squandles in turn sim...

We show that in every codimension greater than one there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) which cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word ‘...