Mathematics

Let p be a prime, n≥1 , K(n) the n th Morava K -theory spectrum, G n the extended Morava stabilizer group, and K(A) the algebraic K -theory spectrum of a commutative S -algebra A . For a type n+1 complex V n , Ausoni and Rognes conjectured that (a) the unit map i n : L K(n) ( S 0 )→ E n from the K(n) -local sphere to the Lubin-Tate spectrum induces a map K( L K(n) ( S 0 ))∧ v −1 n+1 V n →(K( E n ) ) h G n ∧ v −1 n+1 V n that is a weak equivalence, where (b) since G n is profinite, (K( E n ) ) h G n denotes a continuous homotopy fixed point spectrum, and (c) π ∗ (−) of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For n=1 , p≥5 , and V 1 =V(1) , we give a way to realize the above map and (c), by proving that i 1 induces a map K( L K(1) ( S 0 ))∧ v −1 2 V 1 →(K( E 1 )∧ v −1 2 V 1 ) h G 1 , where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence (K( E 1 )∧ v −1 2 V 1 ) h G 1 ≃(K( E 1 ) ) h ˜ G 1 ∧ v −1 2 V 1 , where (K( E 1 ) ) h ˜ G 1 is the homotopy fixed points with G 1 regarded as a discrete group.

In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is a quasicategory has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and show that the canonical functor from the fundamental category of a simplicial set to the fundamental category of its realisation is essentially surjective. We show that such a functor sends split monomorphisms to isomorphisms, in particular we show that it is not necessarily an equivalence of categories. On the other hand, we show that the fundamental category of the realisation of the simplicial circle is equivalent to the monoid of the natural numbers. To conclude, we define left covers of locally stratified spaces and we show that, under suitable assumptions, the category of representations of the fundamental category of a simplicial set is equivalent to the category of left covers over its realisation.

In this paper we prove a new formula for the coefficients of the cellular homology of real flag manifolds in terms of the height of certain roots. In particular, for flag manifolds of type A, we get a very simple formula for these coefficients and an explicit expression for the first and second homology groups with integer coefficients.

We propose a new model for the theory of (∞,n) -categories (including the case n=∞ ) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.

We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. In order to do so, we introduce a homology theory with coefficients in a semiring: by explicitly removing additive inverses, we are able to detect directed cycles algebraically. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach.

We construct a nerve from double categories into double (∞,1) -categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double (∞,1) -categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2 -categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2 -categories and the model structure for 2 -fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2 -fold complete Segal spaces.

Let p>3 be a prime number and let r be an integer with 1<r<p-1. For each r, let moreover G_r denote the unique quotient of the maximal class pro-p group of size p^{r+1}. We show that the mod-p cohomology ring of G_r has depth one and that, in turn, it satisfies the equalities in Carlson's depth conjecture [3].

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups). (Some minor mistakes are corrected)

In this paper, we use the minimal categorical background and maximal concreteness to study equivariant factorization homology in the V -framed case. We work with a finite group G and an n -dimensional orthogonal G -representation V . We set up a monadic bar construction for the equivariant factorization homology for a V -framed manifold with coefficients in E V -algebra. We then prove the nonabelian Poincaré duality theorem using a geometrically-seen scanning map.