Papers

We explicitly compute and tabulate the suspension map for cohomology operations, applying it to the dimensional reduction of anomalies and higher group structures of generalised symmetries in quantum field theories and lattice systems. In a quantum system with a discrete $p$-form symmetry, we can restrict the symmetry defects to a codimension-$q$ subspace of spacetime, or equivalently confine the gauge field to a slab whose thickness approaches zero. This results in a $(p-q)$-form symmetry on the subspace. We determine the fate of two important properties of symmetries under this process. First, we compute the anomaly of the reduced symmetry, or the higher anomaly of the original symmetry, which acts as an obstruction to higher gauging and onsiteability. Second, for two symmetries forming a higher-group structure, encoded as a Postnikov class, we compute the reduced higher-group structure, which may or may not be trivial. Both cases are captured by the $q$-fold iteration of the cohomology suspension $\Omega$, also known as the loop functor or transgression, in the cohomology of Eilenberg–MacLane spaces. Classical results state that $\Omega$ is an isomorphism in a stable range and annihilates mixed anomalies. To calculate $\Omega$ over a base ring which is not a field, we make use of small models for the chain DGAs of Eilenberg–MacLane spaces, pioneered by Cartan and Moore, whose results we review and extend. The critical added step is the improvement of a mod $p$ chain contraction to the $p$-local integers, obtained via a $p$-adic series. We automate the computations in a software package emcm, and provide extensive tables of the results. Finally, to make contact with explicit cochain formulations of cohomology operations, we demonstrate how to calculate cup products and certain cup-$i$ products in this framework.

A continuous surjection $f: X \rightarrow Y$ is an almost covering map if there exists a nowhere dense closed set $R \subset Y$ with a finite decomposition $R = \bigsqcup_{\beta} T_{\beta}$ such that the restriction $f\vert_{f^{-1}(Y \setminus R)}$ is a covering map and, for each $\beta$, the restriction $f\vert_{f^{-1}(T_{\beta})}$ is a fiber bundle. We refer to an almost covering map as a topological semi-small map if, for each $\beta$, the fiber of $f\vert_{f^{-1}(T_\beta)}$ and the space $T_\beta$ satisfy the same dimension condition as for algebraic semi-small maps. We first study the topological properties of almost coverings and develop the necessary tools. In the next step, we restrict our attention to topological semi-small maps and prove that, under the assumption that for each $\beta$ and each trivializing neighborhood $U \subset T_{\beta}$ the connecting homomorphisms in the Gysin sequence of the normal bundle of $f^{-1}(U) \subset X$ vanish in suitable degrees, the derived direct image of the constant sheaf on the domain admits a decomposition.

The cohomology of a differentiable stack can be modelled by the Bott-Shulman-Stasheff bicomplex. However, computing its cohomology is still difficult in many examples. Improving on this requires additional methods such as using spectral sequences and finding models for the cohomology of each column of the bicomplex. This paper provides a first attempt of relating and unifying two approaches of this form. Firstly, Arias Abad has generalised the Bott spectral sequence to Lie groupoids by introducing representations up to homotopy. The second approach is the recent work on a model for the cohomology of differentiable stacks associated to proper and regular Lie groupoids by this author.

We construct a smooth, equivariantly formal, simply connected torus $8$-manifold over a contractible $4$-manifold with the boundary a Brieskorn homology $3$-sphere having a nontrivial fundamental group. We prove extension to a torus graph for GKM graphs of GKM$_{4}$ manifolds in complexity $2$, and $3$ under a finite fundamental group assumption (that is not satisfied, in general).

Free circle action on manifolds has been explored in several articles. Gonzalez and Velasco considered some free circle actions on the finite product of spheres. In this paper, we introduce generalized complex projective product spaces, extending their definition and the concept of Dold manifolds. This produces infinitely many different classes of new smooth manifolds. First, we study the integral cohomology rings and stable tangent bundles on certain generalized complex-projective product spaces. Then, we discuss the product inequality for strongly equivaraint TC. We exhibit the closeness of the lower and upper bounds for the LS-category and topological complexity of several classes of generalized complex projective product spaces. In many cases, we compute the exact value of the LS-category.

We develop simplicial-set models for dagger $(\infty,1)$-categories. We first establish a model structure on dagger simplicial categories whose weak equivalences are local weak equivalences that are essentially surjective up to coherently unitary equivalence. We then construct the dagger Joyal model structure on dagger simplicial sets and prove that the dagger rigidification–nerve adjunction is a Quillen equivalence. We next introduce the unitary core and use it to characterize dagger Joyal equivalences intrinsically. Finally, we compare the dagger Joyal model structure with the DCH anti-involutive Joyal model structure.

Persistent homology summarizes a filtration by recording when homological features appear and disappear, but the resulting barcode does not determine a preferred homology basis or preferred cycle representatives. We study the additional information carried by a basis selected through matrix reduction and introduce the persistence pairing graph, a reduction-dependent directed graph that records how selected homology classes are re-expressed when their associated persistence intervals end. After fixing a filtered finite CW complex, a coefficient field, a filtration-compatible cell order, and a reduced factorization of the boundary matrix, the reduction determines selected birth cycles. When an associated persistence interval ends, the image of its selected birth class has unique coordinates in the basis formed by selected classes whose intervals survive beyond the same filtration value, and the nonzero coordinates define the graph edges. These coordinates can also be recovered through the Kronecker pairing with a dual cohomology basis. We prove that the active selected birth cycles form a homology basis at every cellwise filtration stage, characterize the kernel and image of each homological transition, and show that the graph is directed and acyclic, with every edge pointing toward an interval containing the source interval. In dimension zero, under the standard reduction convention, the graph agrees with the elder rule merge tree. We also prove that its edges can change while the persistence pairs and barcode remain fixed. Finally, we define a bounded fused graph discrepancy and separate the contribution controlled by ordinary persistence stability from the reduction-dependent graph structure. The graph is therefore a descriptor relative to a fixed reduction convention, rather than an invariant of the persistence module alone.

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.

We prove two structural results about factorization systems on $\infty$-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of $\infty$-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for $\infty$-categories equipped with a factorization system, using Juran's double $\infty$-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of $n$-uple $\infty$-categories. Secondly, we prove that the forgetful functor from $\infty$-categories equipped with a factorization system to $\infty$-categories is a monadic right adjoint.

We prove a conjecture proposed by Dwyer and Hess (2014), namely that the category of symmetric sequences, when equipped with the composition and matrix monoidal structures, forms a 2-monoidal category in the sense of Aguiar and Mahajan (2010). Our approach leverages a detailed analysis of the interplay between these two monoidal structures, offering new insights into their behavior. As a consequence, we are able to show that every operad admits a Day-operad structure.

We define the notion of weighted polyhedron which can be thought of as the geometric realization of a weighted simplicial complex introduced by Dawson. Moreover, we will define a weighted version of singular homology theory for a weighted polyhedron and prove that it is isomorphic to the weighted simplicial homology of the weighted polyhedron. This implies that weighted simplicial homology is an invariant under isomorphisms and more generally under certain type of homotopy equivalences of weighted polyhedra. Moreover, we will generalize the cup product and cap product to weighted singular cohomology. In addition, we will interpret some known theories of orbifolds in terms of our weighted singular homology and cohomology.

For a vertex operator algebra $V$ and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $\Sigma$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $\Omega_V(\Sigma;-)$ of the mapping class group of $\Sigma$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $\Omega_V(\Sigma)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $\Omega_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $\Omega_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.

We introduce a framework of (Lie-)global $\infty$-categories, which formalizes various families of $\infty$-categories indexed by compact Lie groups and equipped with suitable restriction functors along continuous group homomorphisms that occur naturally in equivariant homotopy theory and representation theory. As our main results, we show that in this framework unstable and stable equivariant and global homotopy theory admit universal properties, refining and generalizing the results for finite groups from arXiv:2301.08240 and arXiv:2307.11001. In particular, we characterize the passage from unstable to stable equivariant and global homotopy theory at the level of global $\infty$-categories as universally inverting the action of representation spheres in an appropriate sense. Building on this, we define parametrized equivariant and global Thom spectrum functors and show that they recover classical Thom spectrum constructions defined in terms of pointset models.

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty,\infty)$-categories, it represents a concrete and explicit example of a right $(\infty,\infty)$-category that is itself non-contractible, but whose reflection to a left $(\infty,\infty)$-category is contractible.

This paper presents a complete classification of the formality of configuration spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$. We define a constructible de Rham-Dolbeault cohomology theory which provides a constructible CDGA (commutative differential graded algebra) model of $\Conf_m(\mathbb{R}^{n'} \times \mathbb{C}^{n})$. For $(n'=0,n\ge2)$ or $(n'=1,n\ge1)$, the CDGAs are non-formal. For $n'\ge2,n\ge1$, we establish an explicit quasi-isomorphism between the constructible CDGA and its cohomology by using a diagrammatic CDGA of admissible diagrams and a regularized configuration space integral, which leads to the formality. As an application, we show that the local operator algebra of a topological-holomorphic field theory on $\mathbb{R}^{n'} \times \mathbb{C}^{n}$ ($n'\ge2,n\ge1$) is homotopically equivalent to a higher dimensional analog of vertex algebras.

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

We prove an $(\infty,2)$-categorical version of Koszul duality for operads and cooperads in spectra by showing that there is an equivalence between a Morita category of operads and bimodules and a dual Morita category of cooperads and bicomodules. This result subsumes various forms of operadic Koszul duality present in the literature.

We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of $n$-point metric spaces where the growth of the doubling dimension is in $o(\log n)$ and the diameter is bounded. In the $1$-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive $\varepsilon$-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in $n$. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris–Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a lune, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its cost is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin–Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincaré duality of the rational (non-equivariant) graph cohomology algebra. As an application, we prove that orientable GKM graphs remain connected after removing any single vertex.

The Andreadakis-Johnson filtration and its associated construction, known as the Johnson homomorphism, are useful tools in group theory. They provide a step-by-step approach to studying the automorphisms of a given group. After explaining the basics we survey both classical and recent results around the Andreadakis-Johnson filtration, with emphasis on the mapping class group of a once-bordered surface.

Given an object $A$ in a big tensor-triangulated category, we study the homological and cohomological Bousfield classes of the associated localization: the tensor-triangulated category of $A$-local objects. We show that the homological support classifies the homological Bousfield classes of the $A$-local category precisely when an $A$-relative form of the homological detection property holds. Moreover, we prove that this holds if and only if $A$ is Bousfield equivalent to a coproduct of homological residue fields. The analogous classification of cohomological Bousfield classes by homological cosupport is strictly stronger: it is equivalent to an $A$-relative form of homological stratification. This equivalence between stratification and the classification of cohomological Bousfield classes is new even in the absolute case. A further surprise is that stratification is also equivalent to the classification of homological Bousfield classes together with the statement that every cohomological Bousfield class is homological. Applied to chromatic homotopy theory, these results classify the homological Bousfield classes of any localization of spectra with respect to a coproduct of Morava $K$-theories. This covers many localizations of interest. We also completely characterize when such chromatic localizations are relatively homologically stratified. This yields new examples of cohomological Bousfield classes that are not homological. In particular, it answers a question of Wolcott concerning the category of harmonic spectra. Our examples are produced by exhibiting local spectra with empty homological cosupport.

This paper relates algebraic K-theory of fields to polylogarithms via general linear groups. We focus on the case of number fields and prove that the motivic realisation map from the Goncharov Lie coalgebra to the motivic Lie coalgebra is an isomorphism. This implies the Goncharov universality conjecture and a structural result for special values of Dedekind zeta functions. We also construct explicit polylogarithmic cocycles representing nonzero multiples of the Borel classes.

Let \(D\in\{\C,\Hh,\Oct\}\) be a real composition division algebra of dimension \(d\in\{2,4,8\}\). We construct an explicit rank-\(d\) real vector bundle $ E_D\longrightarrow\RP^d$ by deleting one homogeneous coordinate on each standard affine chart, interpreting the remaining coordinates as an element of \(D\), and using the corresponding left-multiplication matrices as transition data. The bundle admits a matrix-determined trivialization away from the \(d+1\) coordinate points. Relative to this trivialization, each deleted point has local clutching map $S^{d-1}\longrightarrow\SO(d),\; u\longmapsto L_u$, which is respectively the complex, quaternionic, or octonionic Hopf clutching map. A section arising from the same matrices has exactly the coordinate points as nondegenerate zeros. Consequently, $w(E_D)=1+x^d$, and real \(K\)-theory together with Euler-class cancellation gives $E_D\cong\gamma^{\oplus d}$. Thus the construction supplies explicit framed-defect realizations of these familiar bundles. In particular, it gives a semialgebraic octonionic realization of \(\gamma^{\oplus8}\) on \(\RP^8\) with nine specified local Hopf defects. We also describe the associated Pfister quadratic bundle over arbitrary fields.

We present a direct geometric construction which extends the Reshetikhin–Turaev TQFT to circles. We prove that this agrees with the generators-and-relations approach of Bartlett–Douglas–Schommer-Pries–Vicary, thus providing an alternative to the Cerf-theoretic part of their classification of once-extended 3-dimensional TQFTs, and deduce as a result that all once-extended 3-dimensional TQFTs arise via our construction.

In this paper, we prove a product rule for Goodwillie derivatives: given a differentiable $\infty$-category $\mathcal{C}$ whose stabilization is equivalent to the $\infty$-category $\mathrm{Sp}$ of spectra, we show that the derivatives functor $\partial_* \colon \mathrm{Fun}^\omega(\mathcal{C}, \mathrm{Sp}) \to \mathrm{RMod}_{\partial_*\mathrm{id}_\mathcal{C}}(\mathrm{SSeq}(\mathrm{Sp}))$ is strong symmetric monoidal, where the source is equipped with the pointwise tensor product and the target with Day convolution. Since the Koszul dual of $\partial_*\mathrm{id}_\mathcal{C}$ can be recovered as a coendomorphism operad from Day convolution, this product rule is useful for calculating the operad $\partial_*\mathrm{id}_\mathcal{C}$ in examples. We derive the product rule as a consequence of the more general statement that taking derivatives preserves cartesian products on the $(\infty, 2)$-categorical level. In fact, the main theme of this paper is that the extraction of Goodwillie derivatives preserves a lot of structure when regarded as a functor of $(\infty, 2)$-categories: apart from products, it also preserves cotensors and certain pullbacks. We illustrate how our results can be used to calculate Goodwillie derivatives by determining the operad structure on the derivatives of the identity functor in pointed spaces, algebras over an operad and sheaves on a site.

We construct the half-signature bordism bicategory $\mathbf{Bord}_{1,2,3}^\mathrm{sig/2}$ as a central extension of the 3-dimensional oriented bordism bicategory by $\mathbb{Z}$. This is an index $2$ subextension of the signature bordism bicategory. Following methods of Bartlett–Douglas–Schommer-Pries–Vicary, we construct a presentation of this bicategory and hence show that its linear representations are classified exactly by modular tensor categories.

Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let $\text{SL}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The homology groups $H_k(\text{SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ are of interest in Geometric Group Theory, Algebraic Number Theory and Algebraic K-theory. In this dissertation, we study these groups via a spectral sequence derived by the action of $\text{SL}_2(\mathbb{Z}[1/n])$ on the product of the trees $B_p$ associated with the $p$-adic valuation on $\mathbb{Q}$ where $p$ is a prime factor of $n$.

We develop a $\mathbb{Q}_p$-homotopy theory for $p$-complete spaces. To a $p$-complete space $X$, we associate a commutative differential graded algebra over $\mathbb{Q}_p$ by rectifying the $E_\infty$-algebra $S^*(X;\widehat{\mathbb{Z}}_p)\otimes_{\widehat{\mathbb{Z}}_p} \mathbb{Q}_p$ of singular cochains. For nilpotent $p$-complete finite type spaces, we prove that the minimal model of this algebra recovers the $\mathbb{Q}_p$-homotopy groups and Whitehead products, in direct analogy with Sullivan's rational homotopy theory. We also prove that, for a non-simply-connected $p$-complete space, the Lie algebra dual to its $1$-minimal model is the Lie algebra of the continuous Mal'cev $\mathbb{Q}_p$-completion of the fundamental group. We apply the $\mathbb{Q}_p$-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for $p$-complete spaces, finiteness properties of étale homotopy types, formality of smooth proper varieties, Galois representations on étale homotopy groups, and constraints on étale fundamental groups.