Mathematics

Let Emb( S 1 ,M) be the space of smooth embeddings from the circle to a closed manifold M of dimension ≥4 . We study a cosimplicial model of Emb( S 1 ,M) in stable categories, using a spectral version of Poincaré-Lefschetz duality called Atiyah duality. We actually deal with a notion of a comodule instead of the cosimplicial model, and prove a comodule version of the duality. As an application, we introduce a new spectral sequence converging to H ∗ (Emb( S 1 ,M)) for simply connected M and for major coefficient rings. Using this, we compute H ∗ (Emb( S 1 , S k × S l )) in low degrees with some conditions on k , l . We also prove the inclusion Emb( S 1 ,M)→Imm( S 1 ,M) to the immersions induces an isomorphism on π 1 for some simply connected 4 -manifolds, related to a question posed by Arone and Szymik. We also prove an equivalence of singular cochain complex of Emb( S 1 ,M) and a homotopy colimit of chain complexes of a Thom spectrum of a bundle over a comprehensible space. Our key ingredient is a structured version of the duality due to R. Cohen.

We study certain formal group laws equipped with an action of the cyclic group of order a power of 2 . We construct C 2 n -equivariant Real oriented models of Lubin-Tate spectra E h at heights h= 2 n−1 m and give explicit formulas of the C 2 n -action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory M U R , and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of M U R .

We compute the stable homotopy groups up to dimension 90, except for some carefully enumerated uncertainties.

In this paper, we shall compute the chain complex and the corresponding homology of some Morse function f over integer coefficients. The definition of the correct boundary operator requires a careful construction of moduli space of (pseudo)gradient flow lines orientations. We will then apply this construction in the computation of these homology groups on 4-manifolds.

We develop Morse-Bott theory on posets, generalizing both discrete Morse-Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik-Schnirelmann theorem for general matchings on posets, in particular, for Morse-Bott functions.

We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.

In the paper we introduce a general approach how for a given virtual biquandle multi-switch (S,V) on an algebraic system X (from some category) and a given virtual link L construct an algebraic system X S,V (L) (from the same category) which is an invariant of L . As a corollary we introduce a new quandle invariant for virtual links which generalizes previously known quandle invariants for virtual links.

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal G -spectra from symmetric monoidal G -categories. The new machine produces highly structured associative ring and module G -spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal G -categories to the multicategory of orthogonal G -spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of G -spectra in arXiv:1110.3571.

In this paper, we calculate the image of the connective and periodic rational equivariant complex K -theory spectrum in the algebraic model for naive-commutative ring G -spectra given by Barnes, Greenlees and Kędziorek for finite abelian G . Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. We further deduce a structure theorem for module spectra over rational equivariant complex K -theory.

Eilenberg-MacLane spaces, that classify the singular cohomology groups of topological spaces, admit natural constructions in the framework of simplicial sets. The existence of similar spaces for the intersection cohomology groups of a stratified space is a long-standing open problem asked by M. Goresky and R. MacPherson. One feature of this work is a construction of such simplicial sets. From works of R. MacPherson, J. Lurie and others, it is now commonly accepted that the simplicial set of singular simplices associated to a topological space has to be replaced by the simplicial set of singular simplices that respect the stratification. This is encoded in the category of simplicial sets over the nerve of the poset of strata. For each perversity, we define a functor from it, with values in the category of cochain complexes over a commutative ring. This construction is based upon a simplicial blow up and the associated cohomology is the intersection cohomology as it was defined by M. Goresky and R. MacPherson in terms of hypercohomology of Delignes's sheaves. This functor admits an adjoint and we use it to get classifying spaces for intersection cohomology. Natural intersection cohomology operations are understood in terms of intersection cohomology of these classifying spaces. As in the classical case, they form infinite loop spaces. In the last section, we examine the depth one case of stratified spaces with only one singular stratum. We observe that the classifying spaces are Joyal's projective cones over classical Eilenberg-MacLane spaces. We establish some of their properties and conjecture that, for Goresky and MacPherson perversities, all intersection cohomology operations are induced by classical ones.