Papers

We compute the rational homology of $\mathrm{Out}(F_8)$ and $\mathrm{Aut}(F_7)$, and several related groups. We also show that the Eisenstein classes in $H_7(\mathrm{Aut}(F_5))$ and $H_{11}(\mathrm{Aut}(F_7))$ are zero.

Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $\pi _{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic Weyl invariants of $G$ to suitable simple factors of $P$, which also address a key nonvanishing issue not settled by earlier approaches of Meier, Shiga-Tezuka, Kotschick, and Terzic.

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

We show that the cohomology rings of toric Richardson varieties in the Grassmannian are finite truncations of the ring of quasisymmetric functions. We exhibit an affine paving of each such variety whose cell closures give rise to the basis of fundamental quasisymmetric functions. We similarly interpret the homology of these varieties in terms of the ring of noncommutative symmetric functions and show that the expansion of the homological class of any torus-invariant subvariety into the affine paving basis agrees with the expansion of a corresponding generalized noncommutative ribbon function into the ribbon basis. By taking the direct limit of all toric Richardson varieties, we obtain an ind-variety equipped with a weak $H$-group structure whose cohomology is the Hopf algebra of quasisymmetric functions. We conjecture that it is isomorphic to a similar $H$-group constructed by Baker–Richter. As a byproduct, we deduce that the $f$-vectors of shard polytopes are log-concave, making the first progress on a question of Ferroni–Schröter for matroid base polytopes.

We show that for a 1-category C, the ({\omega}, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict ({\omega}, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict {\omega}-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict ({\omega}, k)-categories, building on previous work by the first-named author.

We introduce equivariant persistent analogues of Lusternik-Schnirelmann category, topological complexity, cup length, and zero-divisors cup length for persistent spaces with group actions, and establish their stability. In particular, we investigate their behavior under equivariant interleavings and obtain bounds for the corresponding erosion distances. For Vietoris–Rips filtrations of compact metric spaces equipped with compatible group actions, we relate the erosion distances between the resulting equivariant persistent invariants to appropriate equivariant versions of the Gromov-Hausdorff distance.

In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle $\mathbb{Z}_{n}$. First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.

Given a smooth manifold, a tangential $O\langle n \rangle$-structure is a lift of the classifying map of its tangent bundle to $BO\langle n \rangle$. An $O\langle n \rangle$-manifold is a manifold equipped with an equivalence class of tangential $O\langle n \rangle$-structures. The cases $n=2$ and $n=4$ recover the familiar notions of oriented and spin manifolds. Larger values of $n$ furnish higher analogs of these with additional geometric properties. In this thesis, we calculate the 2-primary component of the cobordism groups of compact $O\langle n\rangle $-manifolds in a range of degrees, for $n$ between 9 and 17. We derive various consequences for the boundaries of almost-closed manifolds.

A formal category theory is constructed (in the form of a proarrow equipment), encoding weak coherent enrichment over a monoidal model category $\mV$. We describe how basic categorical concepts formulated via the equipment translate back to enriched categories. We characterize Dwyer-Kan equivalences of enriched categories as $2$-categorical equivalences. Specializing to either the Kan-Quillen model structure on simplicial sets, or the Quillen-Serre model structure on topological spaces, we prove that the resulting formal category theory is equivalent to the one associated with the $\infty$-cosmos of quasicategories, thereby extending the formal approach to $(\infty,1)$-categories in the sense of Riehl-Verity to encompass both simplicial and topological categories. A notion of classifying object, formulated internally to the equipment of $\mV$-categories, leads to enriched versions of Quillen's Theorem A.

We introduce partition functors: algebraic structures indexed by partitions of finite sets and equipped with restriction and transfer maps along refinements. We construct a monad PD on the category of partition functors and on several categories of partition functors with additional multiplicative structure. For a partition ring, exponential elements in its associated completed ring of symmetric functions acquire canonical logarithms after applying PD. This gives rise to a universal exponential relation between multiplicative and additive power operations. For representation rings this can be used to recover the classical relation between symmetric powers and Adams operations, while for Morava E-theory it can be used to recover Ganter's exponential relation between symmetric powers and Hecke operators. We show that the representation rings of products of symmetric groups form the initial partition ring and that, for symmetric monoidal partition functors, the monad PD is closely related to symmetric invariant tensors and the divided power envelope. We also construct a symmetric monoidal partition ring carrying the universal exponential element, so that its image under PD carries the universal exponential relation.

For finite groups $G$, we discuss the enrichment of equivariant $KK$- and $E$-theory in genuine $G$-spectra.

We study homotopy theories of higher groupoids in specified geometries, distinguishing geometric representability from realization and descent. Ordinary smooth and Banach-open sheaves admit full Brown categories of fibrant objects. For ordinary geometric groupoids, a crossing-axes obstruction motivates an empty-compatible incomplete Brown structure. Its split-Banach version requires a fixed chart-compatible plot site with local kernel-product closure. These results include all finite unique-horn bounds and their union. For represented derived models, finite matching and an outer-prism filtration give the structural Brown calculus. Nuiten's finite-geometric realization theorem yields the finite derived-smooth structure and its localization. Enriched realization base change on the original small Kan-enriched site gives the outer-unbounded hypercomplete structure, with positive-degree geometric horn conditions and Reedy fibrancy in every degree. For Banach domains, structured spectra, strict-open gluing and split-open hyperdescent establish geometric finite limits and hypersheaf representability. They give the represented Brown structure, using the same hypercompleted weak-equivalence class at every outer bound. A square-zero test detects nonordinary self-intersections of the chosen Banach charts, including infinite-dimensional charts. For connective associative affines, homotopy split horns, Pridham's finite effectivity theorem and a pointwise diagonal-fibration argument give categories of fibrant objects for his Artin and Deligne–Mumford classes. The associative theory uses quasi-isomorphisms of algebras and realization in presheaves.

Existing link-based combinatorial preprocessing methods speed up the computation of persistent homology by removing a vertex or edge only when its link remains a cone. We replace this condition with a quantitative homological certificate. The reduced homology of the filtered link of a generator (a vertex or edge) defines a local obstruction module whose future part describes the effect of deleting that generator. Its barcode certifies either exact deletion or an explicit bound on the bottleneck error, and a conflict colouring extends this guarantee to families of generators. Our implementation, HomTrim, removes an additional 13% to 41% of the input edges beyond domination-only preprocessing and reduces backend persistence time by factors ranging from 1.55 to 4.61 on weighted flag filtrations. The same module also yields regularity diagrams that measure how far a generator can move before becoming visible to homology, together with a multiscale stability result.

This paper refines the notion of cyclic Massey products to the bi-graded setting, just as quadruple ABC Massey products refine the notion of quadruple Massey products. The result, we call “cyclic ABC Massey products,” are in general non-trivial and contain information different from the quadruple ABC Massey products.

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

In this paper, we develop a bordism-theoretic approach to rigidity problems for toric and quasitoric manifolds. We prove that two toric manifolds are isomorphic as varieties if and only if they are weakly equivariantly unitary bordant. We also establish a parallel rigidity result for omnioriented quasitoric manifolds satisfying the injectivity condition, showing that their equivariant unitary bordism classes completely determine their omniorientation-preserving equivariant homeomorphism types. Thus, equivariant bordism provides a topological framework for detecting geometric and combinatorial rigidity.

For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\times n$ grid graph $G_{m\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \times n})$. We prove that for $n\ge2$, its reduced homology vanishes in dimensions $i \leq n-2$ and in top dimension, while $\tilde{H}_{n-1}(M(G_{3\times n}))\neq 0$. We also show that $M(G_{3 \times n})$ is simply connected for $n \geq 3$. Consequently, the topological connectivity of $M(G_{3\times n})$ is $n-2$.

The Platonic Representation Hypothesis suggests that increasingly capable models converge toward shared representations. Recent work narrows this claim to shared local neighborhood relationships, finding that capacity-dependent trends in several global similarity measures largely disappear after calibration. We challenge this interpretation by showing that prior local-global comparisons confound structural scale (local versus global) with what is compared: relational structure, defined by which samples are related, versus metric geometry, characterized by quantitative relations such as distances, similarities, or correlations. To disentangle these factors, we construct a controlled $2\times2$ framework that evaluates both relational structure and metric geometry at local and global scales. We introduce $H_0$ skeleton overlap as a global counterpart to mutual $k$-nearest neighbors, together with matched distance-aware variants. Across vision-language models, relational structure exhibits robust representational convergence at both scales after calibration, whereas increasingly stringent distance agreement substantially weakens alignment and progressively flattens the capacity-dependent trend. We further extend the analysis beyond ambient Euclidean geometry by evaluating distance agreement under a Riemannian metric approximation and recover the same structure-geometry pattern. The pattern is also reproduced in video-text representations. Together, these results show that relational convergence extends beyond local neighborhoods to global spanning structure, whereas metric geometry exhibits substantially weaker convergence.

We prove that the real $K$-theory functor is lax symmetric monoidal. Moreover, it is initial among those real functors $\mathsf{Wald}_\infty^\mathrm{gd}\to\mathsf{Sp}^{C_2}$ that are additive and lax symmetric monoidal.

We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous $K$-theory at its center. We generalize the chromatic purity theorem for algebraic $K$-theory to continuous $K$-theory of dualizable stable categories and reformulate it as the purity of $H$-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category $\mathcal{C}$, the $T(n)\oplus T(n-1)$-completion map induces an equivalence $$ K_{T(n)}^{\mathrm{cont}}(\mathcal{C})\xrightarrow{\simeq}K_{T(n)}^{\mathrm{cont}}(\mathrm{Nuc}_{T(n)\oplus T(n-1)}(\mathcal{C})). $$ We also establish chromatic descent for continuous $K$-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the $T(n)$-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type $n$ generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.

Let $\Gamma$ be a coalgebra over a field. There is a well-known covariant embedding of the category of graded $\Gamma$-comodules into the category of graded modules over the dual algebra $\Gamma^*$. We show that this covariant embedding preserves Ext-groups in a range of motivating cases, including in some cases like Steenrod algebras where the covariant embedding has otherwise bad homological properties and fails to preserve Ext-groups on the underlying ungraded categories. As an application, we prove the old conjecture in stable homotopy theory that there are no nonzero projectives in the category of graded comodules over the dual Steenrod algebra. As another application, we carry out a study of the right adjoint to the extended comodule functor, including a topological interpretation of this right adjoint. This yields generalizations of formulas of Margolis and Lin characterizing maps from Eilenberg-Mac Lane spectra into finite-type bounded-below spectra; our generalizations drop the "finite-type" hypothesis.

We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\mathbf{Sh}}(X)$ is homeomorphic to ${\mathrm{Sk}}({\mathrm{Sob}}(X))$, where ${\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\mathbb R}^n$ is homeomorphic to the space of ideals of ${\mathbb R}^n$ with a natural topology.

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of $L^p$-representations. We show that for lattices in simple groups of higher rank, the cohomology with $L^p$-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for $L^1$-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The $L^p$-vanishing below the rank proves a conjecture by Gromov regarding $L^p$-cohomology of symmetric spaces.

We prove a formula for Reidemeister numbers of virtually polycyclic groups that generalises known formulas for finite groups and torsion-free virtually polycyclic groups. Based on this formula, we introduce a new invariant which we call the tamed Reidemeister number, and prove that its associated zeta functions are rational on virtually polycyclic groups. We also show that this implies rationality of (ordinary) Reidemeister zeta functions, as well as rationality of Nielsen zeta functions on compact infra-solvmanifolds.

We prove that no nontrivial finite group $G$ satisfies the equivariant generating hypothesis of Bohmann, even when all $RO(H)$-graded homotopy groups are tested for every subgroup $H\leq G$. For each $G$, we construct ghosts on finite $G$-spectra with arbitrarily long nonzero composition powers. We also prove that the homotopy-module functors are nonfull and construct non-equivalent finite $G$-spectra with isomorphic full homotopy modules. A central ingredient of our proof is the natural ghost operation $\Psi=(p-[C_p/e])(1-g)$ on $C_p$-spectra. Motivated by Ma–Xu's examples in the motivic settings, we apply this operation to both a two-cell complex and products of complex projective spaces with cyclic permutation action, which produces arbitrarily long nonzero ghost powers. We also construct a ghost on an explicit four-cell complex that is detected in the derived category over the group ring.

We present a computer-based approach to computing differentials in the unstable Adams spectral sequence by systematically applying the unstable Leibniz rule and naturality with respect to maps in the EHP sequence. We record our results in tables of upper and lower bounds on the orders of 2-primary unstable homotopy groups of spheres through the unstable 50-stem. We provide examples of proofs for several differentials and give a guide to interpreting the associated unstable Adams charts and flow chart diagrams for differential proofs.

We prove a PL bordism version of Gromov's linearity conjecture over a broad class of asphericalization groups for manifolds endowed with faithful representations in all dimensions. Since every group embeds into an asphericalization group, this bordism version of the conjecture holds after allowing the target group to be enlarged. Our results apply in both the oriented and unoriented settings. In the PL category, no bounded local geometry is required. As an application, we obtain improved linear bounds for the Cheeger–Gromov $L^2$ $\rho$-invariants of PL $(4k-1)$-manifolds associated with their universal covers, replacing the previous superfactorial growth in dimension by factorial growth. The proofs are based on a quantitative asphericalization method that provides explicit control of the complexity of the resulting bordisms.

We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories – such as Atiyah–Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner–Meier – analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$, the categories $\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$ and $\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\mathrm{LS}^\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah–Segal completion theorem. We show that global sections induce an equivalence $\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$ and we prove that $\mathrm{LS}^\mathrm{temp}(X,A)$ is a smashing localization of $\mathrm{LS}^\mathrm{gen}(X,A)$ if $X$ is an orbispace.