A stratified space is a filtered space with manifolds as its strata. Connolly and Vajiac proved an end theorem for stratified spaces, generalizing earlier results of Siebenmann and Quinn. Their main result states that there is a single $K$-theoretical obstruction to completing a tame-ended stratified space. A necessary condition to completeness is to find an exhaustion of the stratified space, i.e. an increasing sequence of stratified spaces with bicollared boundaries, whose union is the original space. In this paper we give an example of a stratified space that is not exhaustible. We also prove that the Connolly-Vajiac end obstructions can be realized.

In this paper, we consider "diagrams of rings", or functors from a small category to the category of rings, and the corresponding diagrams of groups $K_i.$ Classically, this was initiated by Milnor. The main result of this paper is the direct comparison of the filtration in classical algebraic $K$-theory discussed in J. Duflot, "Simplicial groups that are models for algebraic $K$-theory," Manuscripta Math. 113 (2004), no. 4, 423–470 and J. Duflot and C.T. Marak, "A filtration in algebraic $K$-theory," J. Pure Applied Algebra 151 (2000), no. 2, 135–162 to a corresponding filtration in the Bousfield-Kan spectral sequence associated to a Tot-tower of simplicial groups attached to the diagram of rings.

A partial magmatic bialgebra, or (T; S)-magmatic bialgebra, where T ⊂ S are subsets of N ≥ 2, is a vector space endowed with an n-ary operation for each n ∈ S and an m-ary co-operation for each m ∈ T satisfying some compatibility and unitary relations. We prove an analogue of the Poincaré-Birkhoff-Witt theorem for these partial magmatic bialgebras.

We consider a localization construction which appears, at least conjecturally, in symplectic geometry. Besides the general algebraic theory, we also look at a specific example, coming from mirror symmetry.

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with a few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of super complexes, or the derived category of mixed complexes.

We remark that a very short proof of an elementary result concerning cohomological periods is implicit in the existing literature. As a corollary we show that given a finitely generated stably free resolution of the integers over a finite group, two of its modules are free.

We study a correspondence between orientation-reversing involutions on compact 3-manifolds with only isolated fixed points and self-dual, binary codes. We show in particular that every such code can be obtained from such an involution. We further relate doubly even codes to Pin-structures and Spin-manifolds.

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal's model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two different approaches to invertible complete Segal spaces and Segal groupoids.

For $n = 2{m+1} - 4, m \geq 2$, we determine the cup-length of $H^*(\tilde{G}_{n,3}; \mathbb{Z}/2)$ by finding a Gröbner basis associated with a certain subring, where $\tilde{G}_{n,3}$ is the oriented Grassmann manifold $SO(n + 3)/SO(n) \times SO(3)$. As an application, we provide not only a lower but also an upper bound for the LS-category of $\tilde{G}_{n,3}$. We also study the immersion problem of $\tilde{G}_{n,3}$.

An algebraic version of a theorem of Quillen is proved. More precisely, for a regular Noetherian scheme $S$ of finite Krull dimension, we consider the motivic stable homotopy category SH($S$) of $\mathbb{P}^1$-spectra, equipped with the symmetric monoidal structure described in [7]. The algebraic cobordism $\mathbb{P}^1$-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation$th^{MGL} \in \rm{MGL}^{2,1}(\rm{Th}(\mathcal{O}(-1)))$. For a commutative monoid $E$ in the category SH($S$), it is proved that the assignment $\varphi \longmapsto \varphi(th^{\rm{MGL}})$ identifies the set of monoid homomorphisms $\varphi : \rm {MGL} \longmapsto E$ in the motivic stable homotopy category SH($S$) with the set of all orientations of $E$. This result generalizes a result of G. Vezzosi in [12].

We describe a new sequence of polytopes which characterize A∞-maps from a topological monoid to an A∞-space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parametrize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of composihedra, that is, the nth composihedron $CK(n)$.

We prove a new localization theorem for algebraic stable categories when the localizing subcategory is generated by a precovering class in the model category. We use this to show how one may explicitly realize certain Bousfield localization functors that arise naturally in the study of relative homological algebra of groups.

We redefine the cup products in Hopf cyclic cohomology. These cup products were first defined by the author and M. Khalkhali via a relatively complicated method as a generalization of Connes' cup product for cyclic cohomology of algebras. In this paper we use the generalized Eilenberg-Zilber theorem and define the cup product using a bicocyclic module naturally associated to the cocyclic modules of the coalgebras and the algebras in question. In the last part of the paper we derive some formulas for the cup products.