in this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  we relate the cohomology of a normal subgroup $n$ of a topological group $g$ and the quotient $g/n$ to the cohomology of $g$. we get the inflation-restriction exact sequence. also, we obtain a seven-term exact cohomology sequence up to dimension 2. we give an interpretation of the first non-a...

Homotopy groups of a connective ring spectrum R form an Ngraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R repres...

We provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as plethory, originally due Tall and Wraith. Stacey Whitehouse showed this is present on cohomology operations for suitable generalised theory. compute explicit expression plethory complex topological K-theory. This formulated in terms enhanced with corresponding looping operations. In context...

Suppose that F is a number field, with ring of integers OF . Let ` denote an odd prime and let R = OF [1/`]. In [3], the author and W.Dwyer gave an explicit conjectural computation of the mod ` cohomology of the infinite general linear group GLR. Here is the quickest and simplest statement of the conjecture (all homology and cohomology groups have Z/` coefficients): let U denote the infinite un...

Let $A_1$, $A_2$ be unital Banach algebras and $X$ be an $A_1$-$A_2$- module. Applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $A_i$ into the dual space $A^*_i$ (for$i=1,2$) and such derivations  from  the triangular Banach algebraof t...

The main goal of this paper is to associate to each smooth quasi-projective scheme X over any field an E∞ differential graded algebra whose cohomology groups are the higher Chow groups of X. Modulo torsion, this provides a strictly commutative differential graded algebra. Such a construction is currently known only for the case X itself is a field. Needless to say there are several applications...

I Lewin, J. C., Science, 112, 652 (1950). 2 In the experiments reported here, cultures were grown on minimal medium plus sodium acetate (0.2 gm per 100 ml). The minimal medium, described in Levine, R. P., and Ebersold, W. T., Zeit. Vererbungs., 89, 632 (1958), was altered by a fivefold increase in the concentration of Beijerinck's solution. 3 Arnon, D. I., Plant Physiol., 24, 1 (1949). 4The aut...

Boardman, Johnson, and Wilson (1995) gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complex K-theory by taking into account its periodicity, we prove that an unstable algebra for complex K-theory is precisely a filtered λ-ring and vice versa. 1. Introduction. Lambda operations in complex K-theory were f...

In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and theory representations. second main theme at intersection toric topology, homotopy polyhedral products, homology local rings, Stanley–Reisner rings simplicial complexes.

In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...