Mathematics

The problem of the description of the orbit space X n = G n,2 / T n for the standard action of the torus T n on a complex Grassmann manifold G n,2 is widely known and it appears in diversity of mathematical questions. A point x∈ X n is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of X n is introduced which opened the new approach to this problem. It is showed that for n>4 the set of critical points Crit X n belongs to our set of singular points Sing X n , while the case n=4 is somewhat special for which Sing X 4 ⊂Crit X 4 , but there are critical points which are not singular. The central result of this paper is the construction of the smooth manifold U n with corners, dim U n =dim X n and an explicit description of the projection p n : U n → X n which in the defined sense resolve all singular points of the space X n . Thus, we obtain the description of the orbit space G n,2 / T n combinatorial structure. Moreover, the T n -action on G n,2 is a seminal example of complexity (n−3) - action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity.

We define an operad in Top, called FM W 2 . The spaces in FM W 2 come with CW decompositions, such that the operad compositions are cellular. In fact, each space in FM W 2 is the realization of a simplicial set. We expect, but do not prove here, that FM W 2 is isomorphic to the 2-dimensional Fulton-MacPherson operad FM 2 . Our construction is connected to the author's work on the symplectic ( A ??,2) -category, and suggests a strategy toward equipping the symplectic cochain complex with the structure of a homotopy Batalin-Vilkoviskiy algebra.

Let R be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on THH(R; Z p ) : it is concentrated in even degrees, generated by powers of the Bökstedt generator σ , generalizing classical Bökstedt periodicity for R= F p . We study an equivariant generalization of the Postnikov filtration, the *regular slice filtration*, on THH(R; Z p ) . The slice filtration is again concentrated in even degrees, generated by RO(T) -graded classes which can loosely be thought of as the *norms* of σ . The slices are expressible as RO(T) -graded suspensions of Mackey functors obtained from the Witt Mackey functor. We obtain a sort of filtration by q -factorials. A key ingredient, which may be of independent interest, is a close connection between the Hill-Yarnall characterization of the slice filtration and Anschütz-le Bras' q -deformation of Legendre's formula.

Operadic tangent cohomology generalizes the existing theories of Harrison cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These are usually non-trivial to compute. We complement the existing computational techniques by producing a spectral sequence that converges to the operadic tangent cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. Finally, we consider relative Sullivan--de Rham models of a fibration p , where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber homotopy equivalences of p .

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner, the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterized in terms of congruences. This is first shown to be a stable problem and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.

We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filtered CW-complexes, providing a nice model for the homotopy category of stratified spaces.

We show that a system of unstable higher Toda brackets can be defined inductively.

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a Z -graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realised as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.

Let G be a finite group and E be an H ∞ -ring G -spectrum. For any G -space X and positive integer m , we give an explicit description of the smallest Mackey ideal J – – in E – – 0 (X×B Σ m ) for which the reduced m th power operation E – – 0 (X)→ E – – 0 (X×B Σ m )/ J – – is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of G× Σ m -Green functors. This theorem also specializes to characterize the appropriate ideal J – – when E is an ultra-commutative global ring spectrum. We give example computations for the sphere spectrum, complex K -theory, and Morava E -theory.

We use a "twisted group algebra" method to constructively adjoin formal radicals α − − √ n , for α a unit in a commutative ring spectrum or an invertible object in a symmetric monoidal ∞ -category. We show that this construction is classified by maps from Eilenberg-Mac Lane objects to the unit spectrum, the Picard spectrum, and the Brauer spectrum.