We compute the Čech cohomology ring of a countable product of infinite projective spaces, and that of an infinite flag manifold. The method of our first result in fact computes the cohomology ring of a countably infinite product of paracompact Hausdorff spaces, under some mild assumptions.

We enumerate the low-dimensional cells in the Voronoi cell complexes attached to the modular groups $\mathit {SL}_N(\mathbb{Z} )$ and $\mathit {GL}_N(\mathbb{Z} )$ for $N=8,9,10,11$, using quotient sublattice techniques for $N=8,9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8(\mathbb{Z} ) = 0$. We deduce from it new knowledge on the Kummer-Vandiver conjecture.

For associative rings with anti-involution several homology theories exist, for instance reflexive homology as studied by Graves and involutive Hochschild homology defined by Fernàndez-València and Giansiracusa. We prove that the corresponding homology groups can be identified with the homotopy groups of an equivariant Loday construction of the one-point compactification of the sign-representation evaluated at the trivial orbit, if we assume that 2 is invertible and if the underlying abelian group of the ring is flat. We also show a relative version where we consider an associative k-algebra with an anti-involution where k is an arbitrary commutative ground ring.

Segre and Verlinde series have been studied in many cases, including virtual geometries of Quot schemes on surfaces and Calabi–Yau 4-folds. Our work is the first to address the equivariant setting for both ${\mathbb{C}}^2$ and ${\mathbb{C}}^4$ by examining higher degree contributions which have no compact analogue.

1. (i) For ${\mathbb{C}}^2$, we work mostly with virtual geometries of Quot schemes. After connecting the equivariant series in degree zero to the existing results of the first author for compact surfaces, we extend the Segre–Verlinde correspondence to all degrees and to the reduced virtual classes. Additionally, we conjecture that there is an equivariant symmetry of Segre series, which was also observed in the compact setting.

2. (ii) For ${\mathbb{C}}^4$, we give further motivation for the definition of the Verlinde series. Based on empirical data andtorsiopn additional structural results, we conjecture that there is an equivariant Segre–Verlinde correspondence and Segre symmetry analogous to the one for ${\mathbb{C}}^2$.

Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge structure (MHS). More precisely, we focus on covers which arise algebraically in the following way: if U is a smooth connected complex algebraic variety and G is a complex semiabelian variety, the pullback of the exponential map by an algebraic morphism $f:U\to G$ yields a covering space $\pi :U^f\to U$ whose group of deck transformations is $\pi _1(G)$. The new MHSs are compatible with Deligne’s MHS on the homology of U through the covering map $\pi $ and satisfy a direct sum decomposition as MHSs into generalized eigenspaces by the action of deck transformations. This provides a vast generalization of the previous results regarding univariable Alexander modules by Geske, Maxim, Wang and the authors in [16, 17]. Lastly, we reduce the problem of whether the first Betti number of the Milnor fiber of a central hyperplane arrangement complement is combinatorial to a question about the Hodge filtration of certain MHSs defined in this paper, providing evidence that the new structures contain interesting information.

La façon la plus simple de faire d’un graphe fini connexe G un système dynamique est de lui donner une polarisation, c’est-à-dire un ordre cyclique des arêtes incidentes à chaque sommet. L’espace de phase $\mathcal {P}(G)$ d’un graphe consiste en toutes les paires $(v,e)$ où v est un sommet et e une arête incidente à v. Elle donne donc la position et le vecteur initiaux. Une telle condition est équivalente à une arête que l’on munit d’une orientation $e_{\mathcal O}$. Avec la polarisation, chaque donnée initiale mène à une marche à gauche en tournant à gauche à chaque sommet rencontré, ou en rebondissant s’il n’y a en ce sommet aucune autre arête. Une marche à gauche est appelée complète si elle couvre toutes les arêtes de G (pas nécessairement dans les deux sens). Nous définissons la valence d’un sommet comme le nombre d’arêtes adjacentes à ce sommet, et la valence d’un graphe comme étant la moyenne des valences de ses sommets. Dans cet article, nous démontrons que si un graphe plongé dans une surface orientée fermée de genre g possède une marche à gauche complète, alors sa valence est d’au plus $1 + \sqrt {6g+1}$. Nous prouvons de plus que ce résultat est optimal pour une infinité de genres g et qu’il est asymptotiquement optimal lorsque $g \to + \infty $. Cela mène à des obstructions pour les plongements de graphes sur une surface. Puisque vérifier si un graphe polarisé possède ou non une marche à gauche complète s’opère en temps au plus $4N$, où N est le nombre d’arêtes (il suffit de le vérifier sur les deux orientations d’une seule arête donnée), cette obstruction est particulièrement efficace. Ce problème trouve sa motivation dans ses conséquences intéressantes sur ce que nous appellerons ici l’ergodicité topologique d’un système conservatif, par exemple un système hamiltonien H en dimension deux où l’existence d’une marche complète à gauche correspond à une orbite du système topologiquement ergodique, donc une orbite qui visite toute la topologie de la surface. Nous nous limitons ici à la dimension $2$, mais une généralisation de cette théorie devrait tenir pour des systèmes hamiltoniens autonomes sur une variété symplectique de dimension arbitraire.

In the present notes, we study a generalization of the Peterson subalgebra to an oriented (generalized) cohomology theory which we call the formal Peterson subalgebra. Observe that by recent results of Zhong the dual of the formal Peterson algebra provides an algebraic model for the oriented cohomology of the affine Grassmannian.

Our first result shows that the centre of the formal affine Demazure algebra (FADA) generates the formal Peterson subalgebra. Our second observation is motivated by the Peterson conjecture. We show that a certain localization of the formal Peterson subalgebra for the extended Dynkin diagram of type $\hat A_1$ provides an algebraic model for “quantum” oriented cohomology of the projective line. Our last result can be viewed as an extension of the previous results on Hopf algebroids of structure algebras of moment graphs to the case of affine root systems. We prove that the dual of the formal Peterson subalgebra (an oriented cohomology of the affine Grassmannian) is the zeroth Hochschild homology of the FADA.

Given a complex analytic family of complex manifolds, we consider canonical Aeppli deformations of $(p,q)$-forms and study its relations to the varying of dimension of the deformed Aeppli cohomology $\dim H^{\bullet ,\bullet }_{A\phi (t)}(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H^{\bullet ,\bullet }_{A\phi (t)}(X)$. As a direct consequence, $\dim H^{p,q}_{A\phi (t)}(X)$ remains constant iff the Bott–Chern deformations of $(n-p,n-q)$-forms and the Aeppli deformations of $(n-p-1,n-q-1)$-forms are canonically unobstructed. Furthermore, the Bott–Chern/Aeppli deformations are shown to be unobstructed if some weak forms of ${ \partial }{ \bar {\partial } }$-lemma is satisfied.

We prove lower bounds on the density of regular minimal cones of dimension less than seven provided the complements of the cones are topologically nontrivial.

We propose a generalised version of configuration spaces defined by disallowing combinations of simultaneous collisions among the n points determined by a family of forbidden partitions. In the case where the underlying space is a finite graph, we construct a cubical complex with the same homology as this configuration space.

We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety and study the corresponding pullback map on (equivariant) cohomology rings.

The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology $\check H^1(\Omega )$ gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of $\check H^1(\Omega )$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega )$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega )$.

For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $\Omega$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then $J$ fixes an element of $\Omega$.

In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan’s decomposition theorem for a general class of large-scale interacting systems.

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor taking values in the category $\mathsf {GPC}$ of groups with a Polish cover (a category first introduced in this work’s predecessor), followed by (ii) a forgetful functor from $\mathsf {GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors to show that a seminal problem in the development of algebraic topology – namely, Borsuk and Eilenberg’s 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslash \Sigma $ to the $2$-sphere – is essentially hyperfinite but not smooth.

Fundamental to our analysis is the fact that the Čech cohomology functors admit two main formulations: a more combinatorial one and a more homotopical formulation as the group $[X,P]$ of homotopy classes of maps from X to a polyhedral $K(G,n)$ space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber’s theorem reconciling the two. In the course of this work, we record definable versions of Urysohn’s Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space $\mathrm {Map}(X,P)$ in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces $\mathrm {Map}(X,P)$, a relation which, together with the more combinatorial incarnation of , embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, $[X,P]$ falls in the category of definable groups, an extension of the category $\mathsf {GPC}$ introduced herein for its regularity properties, which facilitate several of the aforementioned computations.

For a partially multiplicative quandle (PMQ) ${\mathcal {Q}}$ we consider the topological monoid $\mathring {\mathrm {HM}}({\mathcal {Q}})$ of Hurwitz spaces of configurations in the plane with local monodromies in ${\mathcal {Q}}$. We compute the group completion of $\mathring {\mathrm {HM}}({\mathcal {Q}})$: it is the product of the (discrete) enveloping group ${\mathcal {G}}({\mathcal {Q}})$ with a component of the double loop space of the relative Hurwitz space $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$; here $G$ is any group giving rise, together with ${\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that ${\mathcal {Q}}$ is finite and rationally Poincaré and that $G$ is finite, we compute the rational cohomology ring of $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$.

For closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$, we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$. Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$. Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty $-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.

We show that for $n \neq 1,4$, the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

Dirac rings are commutative algebras in the symmetric monoidal category of $\mathbb {Z}$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger $\infty $-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to $\operatorname {MU}$ and $\mathbb {F}_p$ in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.