A fibred category consists of a functor $p: \mathbf{N} \to \mathbf{M}$ between categories $\mathbf{N}$ and $\mathbf{M}$ such that objects of $\mathbf{N}$ may be "pulled back along any arrow of $\mathbf{M}$". Given a fibred category $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the "base category" $\mathbf{M}$, we show that there exists a lifting of the model structure on $\mathbf{M}$ to a model structure on $\mathbf{N}$. We will refer to such a system as a "fibred model category" and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base category $\mathbf{M}$ may be lifted to right homotopic maps in the fibred category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf{N}$ and $\mathbf{M}$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf{M}$ lifts to a Quillen pair on $\mathbf{N}$.

We study the vanishing of cohomology in a triangulated category, in particular vanishing gaps and symmetry.

The classical Wang sequence (for cohomology of fiber bundles over a sphere) is extended to a more generalized setting, given by gluing together two disc bundles over manifolds along their boundaries.

We study the algebraic topology of configuration spaces as interesting objects in their own right and with the goal of constructing invariants for topological manifolds. We calculate the complete Massey product structure for the universal cover of the space of two point configurations in a three-dimensional lens space. We then construct rational homotopy models for these spaces and calculate the rational homotopy groups.

We construct infinitely many chain homotopically distinct algebraic 2-complexes for the Klein bottle group and give various topological applications. We compare our examples to other examples in the literature and address the question of geometric realizability.

We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for finite p-group actions on metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the total space.

We show that the relative algebraic $K$-theory group $K_{2i}(\mathbb{Z}[x, y]/(xy), (x, y))$ is free abelian of rank 1 and that $K_{2i+1}(\mathbb{Z}[x, y]/(xy), (x, y))$ is finite of order $(i!)^2$. We also find the group structure of $K_{2i+1}(\mathbb{Z}[x, y]/(xy), (x, y))$ in low degrees.

Mod 2 cohomology of the Grassmann manifold $G_{2,n}$ is a polynomial algebra modulo a certain well-known ideal. A Groebner basis for this ideal is obtained. Using this basis, some new immersion results for Grassmannians $G_{2,n}$ are established.

We define the fundamental 2-crossed complex $\Omega^\infty(X)$ of a reduced CW-complex $X$ from Ellis’ fundamental squared complex $\rho^\infty(X)$ thereby proving that $\Omega^\infty(X)$ is totally free on the set of cells of $X$. This fundamental 2-crossed complex has very good properties with regard to the geometrical realisation of 2-crossed complex morphisms. After carefully discussing the homotopy theory of totally free 2-crossed complexes, we use $\Omega^\infty(X)$ to give a new proof that the homotopy category of pointed 3-types is equivalent to the homotopy category of 2-crossed modules of groups. We obtain very similar results to the ones given by Baues in the similar context of quadratic modules and quadratic chain complexes.

It is shown that T-spectrum representable cohomology theories on smooth algebraic varieties satisfy normalization condition over nonreal fields. As a consequence, one can see that the rigidity property holds for all representable theories over considered fields.

We prove the topological invariance of the combinatorial Euler characteristic with the help of a canonical, topologically defined stratification of tame spaces by locally compact, tame strata.

We give a construction of an $L_\infty$ map from any $L_\infty$ algebra into its truncated Chevalley-Eilenberg complex as well as its cyclic and $A_\infty$ analogues. This map fits with the inclusion into the full Chevalley-Eilenberg complex (or its respective analogues) to form a homotopy fiber sequence of $L_\infty$ algebras. Applications to deformation theory and graph homology are given. We employ the machinery of Maurer-Cartan functors in $L_\infty$ and $A_\infty$ algebras and associated twistings which should be of independent interest.

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in "Support varieties and representation type of small quantum groups," Internat. Math. Res. Notices 2010, no. 7, 1346–1362. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.

We give a framework for comparing on the one hand theories of $n$-categories that are weakly enriched operadically, and on the other hand $n$-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by a $\mathcal{V}$-operad $P$. We define weak $n$-categories by iterated weak enrichment using a series of parametrising operads $P_i$. We then show how to construct from such a theory an $n$-dimensional globular operad for each $n \geqslant 0$ whose algebras are precisely the weak $n$-categories, and we show that the resulting globular operad is contractible precisely when the operads $P_i$ are contractible. We then show how the globular operad associated with Trimble’s topological definition is related to the globular operad used by Batanin to define fundamental $n$-groupoids of spaces.

On the tensor product of two homotopy Gerstenhaber algebras we construct a Hirsch algebra structure which extends the canonical dg algebra structure. Our result applies more generally to tensor products of “level 3 Hirsch algebras” and also to the Mayer–Vietoris double complex.

We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\mathcal{SH}$ are strict modules over Voevodsky’s algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\mathcal{SH}$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we deduce that with rational coefficients the slices are in fact motives in the sense of Cisinski-Déglise and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky.

In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $\mathscr{S}_g(K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $\mathscr{S}_{g,n}(K)$, was studied by the authors in a previous paper. The study there relied on stability results for the homology of mapping class groups, $\Gamma_{g,n}$ with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when $n$, the number of boundary components, is positive, or in the closed case when the coefficient modules are trivial. Because of this we present a new proof of the rational homological stability for $\mathscr{S}_g(K)$, that is homotopy theoretic in nature. We also take the opportunity to prove a new stability theorem for closed surfaces in $K$ that have marked points.

In this article we prove that the additive invariant corepresented by the noncommutative motive $\mathbb{Z}[r]$ is the Grothendieck group of endomorphisms functor $K_0\mathrm{End}$. Making use of Almkvist’s foundational work, we then show that the ring $\mathrm{Nat}(K_0\mathrm{End},K_0\mathrm{End})$ of natural transformations (whose multiplication is given by composition) is naturally isomorphic to the direct sum of $\mathbb{Z}$ with the ring $W_0(\mathbb{Z}[r])$ of fractions of polynomials with coefficients in $\mathbb{Z}[r]$ and constant term 1.

We prove that the Voevodsky motive with $\mathbb{Z}$-coefficients (resp. $\mathbb{Q}$-coefficients) of a $\mathbb{Z}$-acylic (resp. $\mathbb{Q}$-acyclic) smooth complex variety of dimension $\leqslant 2$ is isomorphic to that of a point and discuss some related extensions.