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By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are SU(3)/T 2, Sp(2)/T 2, Spin(4)/T², Spin(5)/T² and G2/T². In this paper we calculate the cohomology of the free loop space of the rank 2 complete flag manifolds. [ABSTRACT FROM AUTHOR]

This paper establishes several results for coalgebraic structure in 8-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of En-ring spectra: A B, the associated 8-category of descent data is equivalent to the 8-category of comodules over BA B, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the 8-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors. [ABSTRACT FROM AUTHOR]

We consider a nontrivial action of C2 on the type 1 spectrum Y := M2(1) Λ C(η), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum Y... can also be viewed as the complex points of a finite ℝ-motivic spectrum Yℝ. In this paper, we show that one of the 1-periodic v1-self-maps of Y can be lifted to a self-map of Y... as well as Yℝ. Further, the cofiber of the self-map of Yℝ is a realization of the subalgebra Aℝ (1) of the ℝ-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of Y.... [ABSTRACT FROM AUTHOR]

Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyper- octahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let Mn be the extra-special p-group of order p2n+1(n ⩾ 1) and exponent p². We completely compute the mod p Margolis homology of the image ImRes(A,Mn) for every maximal elementary abelian p-subgroup A of Mn. [ABSTRACT FROM AUTHOR]

We work over the finite field Fq. We introduce a notion of unstable P-algebra over an operad P. We show that the unstable P-algebra freely generated by an unstable module is itself a free P-algebra under suitable conditions. We introduce a family of 'q-level' operads which allows us to identify unstable modules studied by Brown-Gitler, Miller and Carlsson in terms of free unstable q-level algebras. [ABSTRACT FROM AUTHOR]

We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product. [ABSTRACT FROM AUTHOR]

We describe graded commutative Gorenstein algebras Σn(p) over a field of prime characteristic p, and we conjecture that Ext•Verpn+1 (1, 1) ∼=Σn(p), where Verpn+1 are the new symmetric tensor categories recently constructed by the current authors, with Ostrik, and also by Coulembier. We investigate the combinatorics of these algebras, and the relationship with Minc's partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of n. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in Σn(p) with a homogeneous system of parameters in Ext•Verpn+1 (1, 1). These parameters have degrees 2i - 1 if p = 2 and 2(pi - 1) if p is odd, for 1 ⩽ i ⩽ n. This at least shows that Ext•Verpn+1 (1, 1) is a finitely generated graded commutative algebra with the same Krull dimension as Σn(p). For p = 2 we also show that Ext•Ver2n+1 (1, 1) has the expected rank 2n(n-1)/2 as a module over the subalgebra of parameters. [ABSTRACT FROM AUTHOR]