We prove a 3 x 3 lemma for regular Goursat categories.

In this article we study a generalization of the notion of Pfister neighbors. An anisotropic quadratic form $\phi$ over a field $F$ of characteristic not $2$ is called a quasi-Pfister neighbor when the anisotropic part $(\phi_{F(\phi)})_{\an}$ is $F(\phi)$-similar to an $F$-quadratic form $\psi$ where $F(\phi)$ denotes the function field of the projective quadric given by $\phi$. We prove the uniqueness of $\psi$ up to $F$-similarity for forms $\phi$ of dimension $\leq 8$, odd dimension and many others of large dimension, and in these cases we give a precise description of $\psi$.

Let $F$ be a field and $p$ a prime number. The $p$-symbol length of $F$, denoted by $\lambda_p(F)$, is the least integer $l$ such that every element of the group $K_2 F/p K_2F$ can be written as a sum of $\leq l$ symbols (with the convention that $\lambda_p(F)=\infty$ if no such integer exists). In this article, we obtain an upper bound for $\lambda_p(F)$ in the case where the group $F^\times/{F^\times}^p$ is finite of order $p^m$. This bound is $\lambda_p(F)\leq \frac{m}{2}$, except for the case where $p=2$ and $F$ is real, when the bound is $\lambda_2(F)\leq \frac{m+1}{2}$. We further give examples showing that these bounds are sharp.

We define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We apply this to describe the Bredon-homological dimension of nilpotent groups in terms of the Hirsch-rank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension.

The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the non-abelian homology of parity quasi-complexes.

Parity contracting homotopies are defined, determining the parity free resolutions in a canonical way, enabling the nonabelian bar construction.

In this way, the even/odd grouping of the simplicial maps in the cocycle conditions of nonabelian cohomology is explained.

Let $\mathfrak{g}$ be a simple Lie algebra, $V$ an irreducible $\mathfrak{g}$-module, $W$ the Weyl group and $\mathfrak{b}$ the Borel subalgebra of $\mathfrak{g}, \mathfrak{n} = [\mathfrak{b}, \mathfrak{b}], \mathfrak{h}$ the Cartan subalgebra of $\mathfrak{g}$. The Borel-Weil-Bott theorem states that the dimension of $H^{i}(\mathfrak{n}; V)$ is equal to the cardinality of the set of elements of length $i$ from $W$. Here a more detailed description of $H^{i}(\mathfrak{n}; V)$ as an $\mathfrak{h}$-module is given in terms of generating functions.

Results of Leger and Luks and Williams who described $H^{i}(\mathfrak{n}; \mathfrak{n})$ for $i\leq 2$ are generalized: $\dim H^{*}(\mathfrak{n}; \Lambda^{*}(\mathfrak{n}))$ and $\dim H^{i}(\mathfrak{n}; \mathfrak{n})$ for $i\leq 3$ are calculated and $\dim H^{i}(\mathfrak{n}; \mathfrak{n})$ as function of $i$ and rank $\mathfrak{g}$ is described for the calssical series.

In this paper we will study deformations of $A_\infty$-algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an $A_\infty$-algebra. We will compute the truncated Hochschild cohomology of odd Moore algebras and classify them up to a unital weak equivalence. We will construct miniversal deformations of particular Moore algebras and relate them to the universal odd and even Moore algebras. Finally we will conclude with an investigation of formal one-parameter deformations of an $A_\infty$-algebra.

Let $R$ be a P.I.D and let $A$ be a dga over $R$. It is well-known that the graded homology modules $H_{\ast }(A)$ and $% Tor_{\ast }^{A}(R,R)$ alone do not suffice (in general) to determine the homotopy type of the dga $A$. J.H. Baues had built a more precise invariant, the "certain" exact sequence of Whitehead associated with $A.$ Whitehead had built it for CW-complexes. In this work we explore this sequence to show how it can be used to classify the homotopy types of $A$.

In this paper, we give a basic application of our Adams-Novikov spectral sequence analogue based on Real cobordism. Concretely, using that technique, we prove restrictions on Greek letter elements which can be permanent cycles or targets of differentials in the geometric chromatic spectral sequence.

We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations.

In this paper, we show that the rational homotopy Lie algebra of classifying spaces for certain types of hyperbolic coformal $2$-cones is not nilpotent.

There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, simplexes and cubes; thus the morphisms between these omega-categories are determined by morphisms between chain complexes. It follows that the entire theory of omega-categories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omega-categories and calculate some internal homomorphism objects.

Let $E=E_{n}$ denote the Morava $E$-theory spectrum, and let $\Gamma$ be the Morava stabilizer group of ring spectrum isomorphisms of $E$. We revisit the isomorphism $\pi_{*}L_{K(n)}(E\smash E)\cong C(\Gamma, E_{*})$ of graded formal Hopf algebroids, and its dual isomorphism $E^{*}E\cong E_{*}[[\Gamma]]$.

In this note we analyze the topology of the spaces of configurations in the euclidian space $\bbr^n$ of all linearly immersed polygonal circles with either fixed lengths for the sides or one side allowed to vary. Specifically, this means that the allowed maps of a $k$-gon $\langle l_1, l_2, \dots, l_k\rangle$ where the $l_i$ are the lengths of the successive sides, are specified by an ordered $k$-tuple of points in $\bbr^n$, $P_1,~P_2, \dots, P_k$ with $d(P_i, P_{i+1}) = l_i$, $1 \le i \le k-1$ and $d(P_k, P_1) = l_k$. The most useful cases are when $n = 2$ or $3$, but there is no added complexity in doing the general case. In all dimensions, we show that the configuration spaces are manifolds built out of unions of specific products $(S^{n-1})^H\times I^{(n-1)(k-2 -H)}$, over (specific) common sub-manifolds of the same form or the boundaries of such manifolds. Once the topology is specified, it is indicated how to apply these results to motion planning problems in $\bbr^2$.

Let $M^n$ be a closed, connected $n$-manifold. Let $M^{-\tau}$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $M^{-\tau}$ is homotopy equivalent to the Spanier-Whitehead dual of $M$ with a disjoint basepoint, $M_+$. This dual can be viewed as the function spectrum, $F(M, S)$, where $S$ is the sphere spectrum. $F(M, S)$ has the structure of a commutative, symmetric ring spectrum in the sense of [7], [12] [9]. In this paper we prove that $M^{-\tau}$ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, Spanier-Whitehead, and Atiyah define an equivalence of symmetric ring spectra, $\alpha :$M^{-\tau}$ \to F(M, S)$. We discuss applications of this to Hochschild cohomology representations of the Chas-Sullivan loop product in the homology of the free loop space of $M$.

We define a Galois structure between central extensions and extensions in a Maltsev variety. By using the theory of commutators we introduce double central extensions. We then prove that the covering morphisms relative to this Galois structure are precisely the double central extensions.

The classical HKR-theorem gives an isomorphism between the n-th Hochschild cohomology of smooth algebras and the n-th exterior power of their module of Kähler differentials. Here, we generalize it for simplicial graded commutative objects in "good pairs of categories". We apply this generalization to complex spaces and Noetherian schemes and deduce several theorems on the decomposition of their respective (relative) Hochschild (co)homologies.

In this paper, for a formal, path connected, finite-dimensional CW-complex $X$ of finite type and a $q$-connected space $Y$ of finite type with $q \geq \rm{dim} X$, we determine the necessary and sufficient condition for the rational cohomology algebra $H^*(\mathcal{F}_*(X,Y);\mathbb{Q})$ of the function space $\mathcal{F}_*(X,Y)$ of based maps to be free.

The purpose of this note is to give a description of class 2 representations of the absolute Galois group of a field $k$ of characteristic $0$ which satisfy a certain condition of Kummer type. This description is based on Galois cohomology and on the theory of projective representations of finite abelian groups.

We construct an explicit diagonal $\Delta_{P}$ on the permutahedra $P.$ Related diagonals on the multiplihedra $J$ and the associahedra $K$ are induced by Tonks' projection $P\rightarrow K$ [19] and its factorization through $J.$ We introduce the notion of a permutahedral set $% \mathcal{Z}$ and lift $\Delta_{P}$ to a diagonal on $\mathcal{Z}$. We show that the double cobar construction $\Omega^{2}C_{\ast}(X)$ is a permutahedral set; consequently $\Delta_{P}$ lifts to a diagonal on $% \Omega^{2}C_{\ast}(X)$. Finally, we apply the diagonal on $K$ to define the tensor product of $A_{\infty}$-(co)algebras in maximal generality.

We introduce a preordered version of D. Scott's equilogical spaces [29], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper.

This setting is compared with other structures which have been recently used for Directed Algebraic Topology: spaces equipped with an order, or a local order, or distinguished paths, or distinguished cubes.

This work is devoted to an interpretation and computation of the first homology groups of the small category given by a rewriting system. It is shown that the elements of the first homology group may be regarded as the equivalence classes of the flows in a graph of the rewriting system. This is applied to calculating the homology groups of asynchronous transition systems and Petri nets. Examples of calculations are given.

Nonequivariantly, covering spaces over a connected (locally nice) space $X$ are in one-to-one correspondence with actions of the fundamental group of $X$ on discrete sets. For nonconnected spaces we consider instead actions of the fundamental groupoid. In this paper we generalize to the equivariant case, showing that we can use either of two possible notions of action of the equivariant fundamental groupoid. We consider both equivariant covering spaces and the more general notion of equivariant homotopy covering spaces.

We show that the class of second order covering maps of simplicial sets in the sense of R. Brown and G. Janelidze is a part of a factorization system for the class of Kan fibrations of simplicial sets.