If $X$ is a cosimplical $E_{n+1}$ space then $\operatorname{Tot}(X)$ is an $E_{n+1}$ space and its mod 2 homology $H_*(\operatorname{Tot}(X))$ has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to $H_*(\operatorname{Tot}(X))$ admits compatible operations. In this paper we give a positive answer to this question.

Let $k$ be a field, and let $G$ be a finite group. By a theorem of D. Benson, H. Krause, and S. Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology $\gamma_G\in H\! H^{3,-1} \hat{H}^*(G)$ with the following property: Given any graded $\hat{H}^*(G)$-module $X$, the image of $\gamma_G$ in $\mathrm{Ext}^{3,-1}_{\hat{H}^*(G)} (X,X)$ is zero if and only if $X$ is isomorphic to a direct summand of $\smash{\hat{H}^*(G,M)}$ for some $kG$-module $M$. In particular, if $\gamma_G=0$ then every module is a direct summand of a realizable $\hat{H}^*(G)$-module.

We prove that the converse of that last statement is not true by studying in detail the case of generalized quaternion groups. Suppose that $k$ is a field of characteristic $2$ and $G$ is generalized quaternion of order $2^n$ with $n\geq 3$. We show that $\gamma_G$ is non-trivial for all $n$, but there is an $\hat{H}^*(G)$-module detecting this non-triviality if and only if $n=3$.

In this paper we focus on preprocessing for persistent homology computations. We adapt some techniques that were successfully used for standard homology computations. The main idea is to reduce the complex prior to generating its boundary matrix, which is costly to store and process. We discuss the following reduction methods: elementary collapses, coreductions (as defined by Mrozek and Batko), and acyclic subspace methods (introduced by Mrozek, Pilarczyk, and Żelazna).

A real Milnor manifold is the non-singular hypersurface of degree $(1; 1)$ in the product of two real projective spaces. These manifolds were introduced by Milnor to give generators for the unoriented cobordism algebra, and they admit free actions by elementary abelian 2-groups. In this paper, we obtain some results on the free 2-rank of symmetry of products of finitely many real Milnor manifolds under the assumption that the induced action on mod 2 cohomology is trivial. Similar results are obtained for complex Milnor manifolds that are defined analogously. Here the free 2-rank of symmetry of a topological space is the maximal rank of an elementary abelian 2-group that acts freely on that space.

A study is made of the action of the fundamental group of a bouquet of a circle and a 1-connected space on the higher homotopy groups. If the 1-connected space is a suspension space, it is shown, with the aid of a theorem of Hartley on wreath products of groups and the Hilton-Milnor theorem, that the action is residually nilpotent. An unsuccessful approach in the case of a general 1-connected space is discussed, as it has some interesting features.

Given a flat connection $\alpha$ on a manifold $M$ with values in a filtered $L_\infty$-algebra $\mathfrak{g}$, we construct a morphism $\mathsf{hol}^{\infty}_\alpha \colon C_\bullet(M) \rightarrow \mathsf{B} \hat{\mathbb{U}}_\infty(\mathfrak{g})$, which generalizes the holonomy map associated to a flat connection with values in a Lie algebra. The construction is based on Gugenheim's $\mathsf{A}_{\infty}$-version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space $\mathbb{R}^d$, and to generalizations of the holonomy representations of braid groups.

Precubical sets are used to model concurrent systems. We introduce the homology graph of a precubical set, which is a directed graph whose nodes are the homology classes of the precubical set. We show that the homology graph is invariant under weak morphisms that are homeomorphisms.

A generalized Postnikov tower (GPT) is defined as a tower of principal fibrations with the classifying maps into generalized Eilenberg–Mac Lane spaces. We study fundamental properties of GPT’s such as their existence, localization and length. We further consider the distribution of torsion in a GPT of a finite complex, motivated by the result of McGibbon and Neisendorfer. We also give an algebraic description of the length of a GPT of a rational space.

Let $Y$ be a real algebraic variety. We are interested in determining the supremum, $\beta(Y)$, of all nonnegative integers $n$ with the following property: For every $n$-dimensional compact connected nonsingular real algebraic variety $X$, every continuous map from $X$ into $Y$ is homotopic to a regular map. We give an upper bound for $\beta(Y)$, based on a construction involving complexification of real algebraic varieties. In some cases, we obtain the exact value of $\beta(Y)$.

We define invariants of unoriented knots and links by enhancing the integral kei counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for a finite kei $X$ using representations of the kei algebra, $\mathbb{Z}_K[X]$, a quotient of the quandle algebra $\mathbb{Z}[X]$ defined by Andruskiewitsch and Graña. We give an example that demonstrates that the enhanced invariant is stronger than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on $\mathbb{Z}_3$ which is not a $\mathbb{Z}_K[X]$-module to detect the non-invertibility of a virtual knot.

We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base—the so-called versal deformation—which induces all other deformations of the given algebra.

Basic homological lemmas well known for modules over rings and, more generally, in the context of abelian categories, have been extended to many other concrete and abstract-categorical contexts by various authors. We propose a new such extension, specifically for commutative monoids and semimodules; these two contexts are equivalent since the forgetful functors from varieties of semimodules to the variety of commutative monoids preserve all limits and colimits.

The connective $ku$-(co)homology of elementary abelian $2$-groups is determined as a functor of the elementary abelian $2$-group, using the action of the Milnor operations $Q_0, Q_1$ on mod $2$ group cohomology, the Atiyah-Segal theorem for $KU$-cohomology, together with an analysis of the functorial structure of the integral group ring; the functorial structure then reduces calculations to the rank 1 case.

These results are used to analyse the local cohomology spectral sequence calculating $ku$-homology, via a functorial version of local duality for Koszul complexes, giving a conceptual explanation of results of Bruner and Greenlees.

In multivariate data analysis dependence beyond pair-wise can be important. With many variables, however, the number of simple summaries of even third-order dependence can be unmanageably large.

“Concurrence topology” is an apparently new method for describing high-order dependence among up to dozens of dichotomous (i.e., binary) variables (e.g., seventh-order dependence in 32 variables). This method generally produces summaries of dependence of manageable size. (But computing time can be lengthy.) For time series, this method can be applied in both the time and Fourier domains.

Write each observation as a vector of 0’s and 1’s. A “concurrence” is a group of variables all labeled “1” in the same observation. The collection of concurrences can be represented as a filtered simplicial complex. Holes in the filtration indicate relatively weak or negative association among the variables. The pattern of the holes in the filtration can be analyzed using persistent homology.

We applied concurrence topology on binarized, resting-state, functional MRI data acquired from patients diagnosed with attention-deficit hyperactivity disorder and from healthy controls. An exploratory analysis finds a number of differences between patients and controls in the topologies of their filtrations, demonstrating that concurrence topology can find in data high-order structure of real-world relevance.

We show that stable derivators, like stable model cate- gories, admit Mayer-Vietoris sequences arising from cocartesian squares. Along the way we characterize homotopy exact squares and give a detection result for colimiting diagrams in derivators. As an application, we show that a derivator is stable if and only if its suspension functor is an equivalence.

The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph $G_B$ of the partially ordered set $B_{S}$ of simplexes of a simplicial complex $S$ has graph homology that is isomorphic to the simplicial homology of $S$. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph $G_{S}$ of $G_B$, whose homologies are isomorphic to the simplicial homologies of $S$.

For any compact Lie group $G$, there are several well-established definitions of a $G$-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal $G$-spectrum for each compact Lie group $G$. We use the framework of enriched indexed categories to make this precise. We also consider equivariant $K$-theory and $\operatorname{Spin}^c$-cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.

Goodwillie’s proof of the Blakers-Massey Theorem for n- cubes relies on a lemma whose proof invokes transversality. The rest of his proof follows from general facts about cubes of spaces and connectivities of maps. We present a purely homotopytheoretic proof of this lemma. The methods are elementary, using a generalization and modification of an argument originally due to Puppe used to prove the Blakers-Massey Theorem for squares.

In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra $H$. We show that the second cohomology group classifies completely weak crossed products, having a common preunit, of $H$ with a commutative left $H$-module algebra $A$.