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In this paper, using homotopy theoretical methods we study the degrees of maps between -connected -dimensional Poincaré complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincaré complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree. As an application of mapping degrees, we consider maps between -connected -dimensional Poincaré complexes with degree , and give a sufficient condition for these to be homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov’s question: when is a map of degree between manifolds a homeomorphism? For low , we classify, up to homotopy, torsion free -connected -dimensional Poincaré complexes. Bibliography: 29 titles.

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal{R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_*(\Omega \mathcal{Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal{R}_K)$, whereas for $H_*(\Omega \mathcal{Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal{Z}_K$., Comment: 16 pages, published version

В работе методы теории гомотопий применяются для изучения отображений между $(n-1)$-связными $(2n+1)$-мерными комплексами Пуанкаре. Получены необходимые и достаточные условия существования отображений заданной степени для таких комплексов Пуанкаре. Эти условия позволяют явно описать все отображения заданной степени с точностью до гомотопии. В качестве приложения степени отображения рассматривается отображение степени $\pm 1$ между $(n-1)$-связными $(2n+1)$-мерными комплексами Пуанкаре и приводится достаточное условие для того, чтобы данное отображение было гомотопической эквивалентностью. Это дает ответ на гомотопический аналог вопроса Новикова о том, когда отображение степени $1$ между многообразиями является гомеоморфизмом. Для малых $n$ дана классификация $(n-1)$-связных $(2n+1)$-мерных комплексов Пуанкаре без кручения с точностью до гомотопии. Библиография: 29 названий.

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

In this paper we study the normalizer decomposition of a compact Lie group $G$ using the information of the fusion system $\mathcal{F}$ of $G$ on a maximal discrete $p$-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of $\mathcal{F}$-centric, $\mathcal{F}$-radical discrete $p$-toral subgroups to the set of conjugacy classes of chains of $p$-centric, $p$-stubborn continuous $p$-toral subgroups. The map is a bijection when $\pi_0(G)$ is a finite $p$-group. We also prove that the classifying space of the normalizer of a chain of discrete $p$-toral subgroups of $G$ is mod $p$ equivalent to the classifying space of the normalizer of the corresponding chain of $p$-toral subgroups., Comment: 18 pages. Version to appear in Women in Topology 2019 Proceedings

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry., Updated in line with the version to appear in Advances in Mathematics

Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [2] of $\Sigma (X, A)^K$ is then $G$-equivariant after suspension. In the case of $\Sigma_m$-polyhedral products, we give criteria on simplicial $\Sigma_m$-complexes which imply representation stability of $\Sigma_m$-representations $\{H_i((X, A)^{K_m})\}$., Comment: To appear in Algebraic & Geometric Topology

This paper aims to find the most general combinatorial conditions under which a moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment-angle complex over a Golod simplicial complex is a co-$H$-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces., Comment: Published in Advances in Mathematics, 2017

We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.

Given an \(\frac{n}{3}\)-neighbourly simplicial complex K on vertex set [n], we show that the moment-angle complex \(\mathcal Z_K\) is a co-H-space if and only if K satisfies a homotopy analogue of the Golod property. This gives a sufficient condition for the integral formality of \(\mathcal Z_K\).

By using the notion of polyhedral products ( X _ , A _ ) K , we recognise the Bestvina–Brady construction [4] as the fundamental group of the homotopy fibre of ( S 1 , ⁎ ) L → S 1 , where L is a flag complex. We generalise their construction by studying the homotopy fibre F of ( S 1 , ⁎ ) L → ( S 1 , ⁎ ) K for an arbitrary simplicial complex L and K an ( m − 1 ) -dimensional simplex. For a particular class of simplicial complexes L, we describe the homology of F, its fixed points, and maximal invariant quotients for coordinate subgroups of Z m . This generalises the work of Leary and Saadetoglu [13] who studied the case when m = 1 .

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jollenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z_K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop space of Z_K is homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime not equal to 2.

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.

James constructed a functorial homotopy decomposition for path-connected, p ointed CW-complexes X. We generalize this to a p-local functorial decomposition of ΣA, where A is any functorial retract of a looped co-H-space. This is used to construct Hopf invariants in a more general context. In addition, when A = ΩY is the loops space of a co-H-space, we show that the wedge summands of ΣΩY further functorially decompose by using an action of an appropriate symmetric group. As a valuable example, we give an application to the theory of quasi-symmetric functions.

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X_1,..., X_n are spaces and CX_i is the cone on X_i, then the polyhedral product determined by K and the pairs (CX_i,X_i) is homotopy equivalent to a wedge of suspensions of smashes of the X_i's. This generalises earlier work of the two authors in the special case where each X_i is a loop space. Connections are made to toric topology, combinatorics, and classical homotopy theory., 27 pages, typos corrected, Section 8 improved

Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\zk)$, as well as a certain Massey products. We show that the LS category closely relates to vanishing of Massey products in $H^*(\zk)$ and through this connection we describe first structural properties of Massey products in moment-angel manifolds. Some of further applications include calculations of the LS category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and $k$-neighbourly complexes, which double as important examples of hyperbolic manifolds., Added characterisation of LS-category of moment angle complexes over two dimensional surfaces; simplified some arguments

We prove two homotopy decomposition theorems for the loops on simply-connected co- H H -spaces, including a generalization of the Hilton-Milnor Theorem. Several examples are given.

Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p − 1. The first construction goes through when l = p − 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give a practical test for when the criterion holds, and use this to give many new examples of finite H-spaces.

In this paper we compute the integral Pontrjagin homology ring of the based loop spaceon some generalised symmetric spaces G/S, where G is a simple compact Lie group Ghaving the torus S as a stationary subgroup. More precisely, we consider generalisedsymmetric spaces which have zero Euler characteristic, that is, for which the torus S is notmaximal in G. Thus this paper can be consider as a natural sequel to paper [5], where theauthors computed the integral Pontrjagin homology of the based loop space of completeflag manifolds of compact simple Lie groups.Generalised symmetric spaces G/H are defined by the condition that their stationarysubgroup H is the fixed point subgroup of a finite order automorphism of th e group G.These spaces consequently admit finite order symmetries mak ing them to have rich ge-ometry and thus attract constant attention of many geometry focused research startingwith [14],[15] until recent once, see for example [1],[2],[7]. Generalised symmetric spacesplayimportantrolein the theoryof homogeneousspaces and, thus, in manyareas of mathe-matics and physics such as representation theory, combinatorics, string topology. In partic-ular complex homogeneous spaces, which are important examples in complex cobordismsand theory of characteristic classes, can be described by complex generalised symmetricspaces.The methods used in this paper are analogous to that of [5]. By [12] all generalisedsymmetric spaces are Cartan pair homogeneous spaces and, thus, formal in the sense ofSullivan. Therefore,startingfromtheirrationalcohomologyalgebrasandusingthe methodof rational homotopy theory together with the Milnor and Moore theorem which expressesthe rational Pontrjagin homology algebra in terms of the universal enveloping algebra ofthe rational homotopy Lie algebra, we compute the rational homology algebras for thebased loop on generalised symmetric spaces G/S. We prove that the integral loop ho-mology groups of these spaces have no torsion which together with the rational computa-tions enables us to determine their integral Pontrjagin ring structure as well. In particular,we calculate the integral Pontrjagin homology ring of the based loop spaces on the fol-lowing generalised symmetric spaces: SU(2n + 1)/T

Let p be an odd prime. Using homotopy decompositions and spherical fibrations, under certain dimensional restrictions, we obtain upper bounds of the p-primary homotopy exponents of some homogeneous spaces such as generalized complex Stiefel manifolds, generalized complex Grassmann manifolds, SU (2n)/Sp (n), E6/F4 and F4/G2 (the latter for p = 2 and p ? 5).

Let G be a simple, compact, simply-connected Lie group localized at an odd prime p. We study the group of homotopy classes of self-maps [G, G] when the rank of G is low and in certain cases describe the set of homotopy classes ofmultiplicative self-maps H[G, G]. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

We study a universal property of $Sp(2)$ in the category of $3$-local homotopy associative, homotopy commutative $H$-spaces. We show that while $Sp(2)$ fails to be universal in the full category, there is a subcategory in which it is universal for its $7$-skeleton.

Let $P^{2n+1}$ be a two-cell complex which is formed by attaching a $(2n+1)$--cell to a $2m$--sphere by a suspension map. We construct a universal space $U$ for $P^{2n+1}$ in the category of homotopy associative, homotopy commutative $H$--spaces. By universal we mean that $U$ is homotopy associative, homotopy commutative, and has the property that any map $f\colon P^{2n+1}\lra Y$ to a homotopy associative, homotopy commutative $H$--space $Y$ extends to a uniquely determined $H$--map $\bar{f}\colon U\lra Y$. We then prove upper and lower bounds of the $H$--homotopy exponent of $U$. In the case of a mod~$p^r$ Moore space $U$ is the homotopy fibre $S^{2n+1}\{p^r\}$ of the $p^r$--power map on $S^{2n+1}$, and we reproduce Neisendorfer's result that $S^{2n+1}\{p^r\}$ is homotopy associative, homotopy commutative and that the $p^r$--power map on $S^{2n+1}\{p^r\}$ is null homotopic., Comment: 12 pages

In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod [math] Moore space for primes [math] and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for [math] are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod [math] Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod [math] Moore space or to an improvement in the known bounds for the exponent of the [math] –torsion in the homotopy groups of spheres.

Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given., This is the version published by Algebraic & Geometric Topology on 29 November 2006

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative H-space extends to a uniquely determined H-map F: U(X)->Y. Developing a method for recognising certain universal spaces, we show the existence of the universal space F_2(n) of a certain three-cell complex L. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the d_1-differential of the EHP-spectral sequence valid in a certain range., 29 pages

Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups of the homotopy classes of maps from the James construction to loop spaces. Applications to homotopy theory appear in a sequel., Comment: This is the version published by Algebraic & Geometric Topology on 29 November 2006

The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish., Comment: 42 pages

There exist spaces BSol(q) which are the classifying spaces of a family of 2-local finite groups based on certain fusion system over the Sylow 2-subgroups of Spin_7(q). In this paper we calculate the cohomology of BSol(q) as an algebra over the Steenrod algebra A_2. We also provide the calculation of the cohomology algebra over A_2 of the finite group of Lie type G_2(q)., Comment: 13 pages

La mostra "Lifting the Curtain", organizzata dalla Polish Modern Art Foundation di Varsavia e inserita fra gli eventi collaterali alla Biennale di Venezia 2014, è stata riproposta a Vienna dalla Austrian Association for Architecure e in quell'occasione è stato pubblicato il catalogo. "Lifting the curtain" è un progetto di ricerca internazionale sullo sviluppo dell’architettura come risultato di complesse interazioni tra produzione materiale e immateriale, dove molti attori costruiscono reti e condividono le conoscenze. Non si focalizza sugli elementi, ma sui meccanismi di produzione. Il contributo dell’Europa centrale alla formazione e alla diffusione della modernità è tutt’ora in gran parte sconosciuto o visto come un episodio marginale nella storiografia del dopoguerra. Questo progetto intende offrire una prospettiva diversa sui momenti cruciali della storia dell’architettura, sottolineando il contributo delle diverse nazioni e delle diverse discipline, che costituiscono i rapporti dell’architettura moderna. La mostra è la prima parte di una ricerca sullo sviluppo dell’architettura come risultato di complesse interazioni tra produzione materiale e immateriale, dove molti attori costruiscono reti e condividono le conoscenze. Non si focalizza sugli elementi, ma sui meccanismi di produzione.