Papers

We investigate the extent to which Whitehead's theorem remains valid for minimal finite models. We show that it fails in this setting, answering negatively a question posed by Barmak. More precisely, for every $n\geq 2$, we construct a weak homotopy equivalence between two $(2n+4)$-point minimal finite models of $S^n\vee S^{n-1}\vee S^{n-1}$ which are not homotopy equivalent. The minimality of these examples follows from a near-extremal classification theorem: if a connected finite space has at most $2n+3$ points and nonzero $n$th homology over a field, then its order complex is homotopy equivalent either to $S^n$ or to $S^n\vee S^k$ for some $1\leq k\leq n$. Finally, we prove a positive Whitehead-type result: under a natural cohomological rigidity hypothesis, every weak homotopy equivalence between minimal finite models is a homotopy equivalence.

For each prime $p$ and integer $n \geq 1$, we construct a $p$-local spectrum $X_n$ with $X_n^{\otimes n} \not\simeq 0$ but $X_n^{\otimes (n+1)} \simeq 0$. This refutes the retract conjecture of Hovey-Palmieri. Our construction also allows us to determine the cardinality of the Bousfield lattice to be $2^{2^{\aleph_0}}$, answering a question of Dwyer-Palmieri.

We first construct the differential refinement of singular homology, verifying that it satisfies the axioms dual to their cohomological counterparts. Then, we define the differential cap product, leading to Poincaré duality. The essential uniqueness of both differential homology and the cap product is proven. Moreover, we construct the relative, non-compact, and local coefficient versions, so that we can state Poincaré and Lefschetz dualities for every smooth manifold with our without boundary. Afterwards, we develop the analogous refinement of any rationally-even homology theory, with the same properties. We also define a "differentiation map" for fibre bundles, dual to the integration map in cohomology. We conclude by sketching some possible applications of this theory in mathematical physics.

We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen–Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.

We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $\Gamma_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $\Gamma_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-Röndigs-Østvær. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $\Gamma_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.

We promote the $n$-birational motivic homotopy category assignment $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, the birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of schemes in $Sm_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.

We calculate the Batalin-Vilkovisky (BV) algebra structure of Rabinowitz loop homology for compact rank one symmetric spaces. As a consequence, we prove that such a space satisfies a natural homological condition called string point invertibility if and only if its Euler characteristic is equal to the characteristic of the coefficient field for loop homology. This implies certain cases of Viterbo's conjecture on a uniform bound on the spectral norm of exact Lagrangian submanifolds in cotangent disk bundles. Furthermore, we prove that whenever a compact rank one symmetric space is string point invertible, the critical levels of its loop homology classes with respect to an arbitrary Riemannian metric satisfy a resonance condition with respect to degrees and a density condition for closed geodesics. This generalizes results of Hingston and Rademacher for spheres to a broader class of compact rank one symmetric spaces.

Vector-valued functions on finite simplicial complexes give rise to multiparameter sublevel set filtrations and corresponding persistent homology. We study the associated inverse problem for multiparameter persistent homology (MPH) of $n$-filters on a fixed finite simplicial complex. We endow both the space of filters and the moduli space of essentially finite persistence modules with stratifications, where strata are given by orbits of natural actions of order-isomorphisms of the unit $n$-cube. The MPH map is then shown to be equivariant and strongly stratified. Over each stratum in the image, the MPH map restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We provide an upper bound on the dimension of the fibers in terms of multigraded Betti numbers, recovering as a special case the known one-parameter bound obtained by Leygonie and Tillmann (2022).

Shimada recently showed that Frobenius–Witt cotangent complex, the animation of Saito's Frobenius–Witt differentials, vanishes on perfectoid rings, thus it serves as a candidate of “absolute” cotangent complex. In this article, we give a pullback description of their arithmetic extension, and as a consequence, we give a direct description of Frobenius–Witt cotangent complex, which leads to generalizations to derived rings and animated pre-log rings. This description allows us to compute Frobenius–Witt cotangent complex of derived $\delta$-rings as well. We also propose a version of Frobenius–Witt cotangent complex relative to derived $\delta$-rings, and establish a vanishing result for prisms, which generalizes vanishing of Frobenius–Witt cotangent complex of perfectoid rings. Finally, independently of previous considerations, we give a regularity criterion via Frobenius–Witt cotangent complex for $p$-local Noetherian (not necessarily local) rings without $F$-finiteness. We also record the flatness of Frobenius–Witt cotangent complex of valuation rings, with essential ideas due to ChatGPT-6 Astra.

We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

We recast known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive sharp upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tibăr, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.

We develop homological Real trace methods, an approach to understanding the continuous mod two Bredon homology of Real topological periodic homology and Real topological negative cyclic homology, generalizing prior work of Bruner and Rognes. We use this new approach to do several computations, including the continuous mod two Bredon homology of the Real topological negative cyclic homology of Real bordism.

Persistent Homology (PH) is an important tool in Topological Data Analysis for point clouds, but can be prohibitively expensive for large datasets. A practical approximation is to compute PH on a smaller subset of representative points, known as landmarks. Existing landmark selection methods mainly address challenges such as outlier contamination rather than restoring the topological features of the full dataset. To select topologically relevant landmarks, we propose a landmark selection method based on Hodge Laplacian. It assigns each point a topological relevance score based on its harmonic participation and selects points with high scores as landmarks. By quantifying each point's participation in global homological structures, our method restores the PH of the original point cloud more accurately than geometric baselines and a method built upon local PH on both synthetic and real-world datasets.

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

Computational paths record the steps of an equality derivation. We give them a topological semantics that distinguishes derivable rewrites from arbitrary homotopies. Coherent representatives pair traces with paths homotopic to their realizations. We compare a topology retaining the entire trace with one observing only endpoints, length, and paths. Quotienting by the declared rewrites gives a groupoid. Multiplication is continuous when composable pairs carry the quotient topology inherited from composable representatives. This topology can differ from the usual subspace topology on pairs of quotient arrows. We characterize when they agree, give compact-Hausdorff and discrete sufficient conditions, and use the Hawaiian earring to exhibit a failure of agreement. The comparison with geometric homotopy classes is injective exactly when the presentation is geometrically complete. Normal-form certificates give a criterion for completeness. In the universal presentation, all paths are primitive steps and all endpoint-fixed homotopies are allowed rewrites; its quotient recovers the quotient-topologized fundamental groupoid. Circle and torus examples recover the classical based-loop classifications by $\mathbb Z$ and $\mathbb Z^2$. A Lean development supports the construction. A focused Lean 4.32.0 result registered in Palomar covers the topology comparison, additive circle and torus classifications, and a conditional Hawaiian-earring obstruction transfer. We distinguish that result from the earlier Lean 4.24.0 development and from the mathematical exposition.

Topological data analysis (TDA) uses topological techniques to extract meaningful shape-based information from complex datasets. Clustering is a central problem in data analysis, and there has been considerable recent interest in understanding how TDA can inform it. Existing approaches either cluster persistence diagrams directly under Wasserstein-type distances, which is computationally expensive, or use vector representations of diagrams. We propose a kernel $k$-means algorithm built on a convex combination of sliced Wasserstein (SW) kernels, one for each homology under consideration. Unlike other vector representations of persistence diagrams, the SW kernel is both stable and discriminative with respect to the $1$-Wasserstein distance. The method outperforms the baselines on two benchmark datasets and remains competitive on a third synthetic dataset, while being computationally efficient. It also outperforms both a single SW kernel on the union of all homology groups and an SW kernel computed directly on the point clouds. The convex combination assigns an interpretable weight to each $q$-homology kernel. We further validate that the weights identify the discriminating homology.

The well-known persistence algorithm summarizes the evolution of homological cycles into what is called a barcode while scanning an input simplicial filtration. We show that this summarization process can be enriched by monitoring other algebraic structures that weave through different dimensions. In particular, we propose an algorithm to monitor the $(p+1)$-chains that make $p$-cycles to be $p$-boundaries and then morph into $(p+1)$-cycles. In effect, we get extra bars called links connecting the bars in dimension $p$ with the bars in dimension $p+1$ in the persistence barcode. The links produce extra barcodes, which we call link barcodes in addition to the usual ones obtained by standard persistence. The link barcodes, as such, are not stable. However, we can make them stable using a fixed “reference” filtration. We apply the link barcodes to the graph isomorphism problem and to the link prediction problem in temporal networks exhibiting its discriminating power through these experiments.

For an odd prime $p$, we show that the homotopy type of a closed $4$-manifold with fundamental group $\mathbb{Z}/p \rtimes \mathbb{Z}$ is determined by its quadratic $2$-type together with an additional $\mathbb{Z}/p$-valued intersection form. We also give a generalization to infinite groups of a result of Hambleton–Kreck on the polarized homotopy classification of Poincaré $4$-complexes, and we prove some partial results on the homotopy classification that are applicable more generally.

The topological Tverberg theorem asserts that if $r$ is a prime power then for any continuous map $f\colon \Delta_{(r-1)(d+1)}\rightarrow \mathbb{R}^d$ from the $(r-1)(d+1)$-dimensional simplex $\Delta_{(r-1)(d+1)}$ to $\mathbb{R}^d$ there exist $r$ pairwise disjoint faces of the simplex whose images have non-empty $r$-fold intersection. By refinement, the same conclusion holds if the simplex is replaced by any polytope of the same dimension. While this dimension is tight for simplices, recent work of Soberón and Zerbib shows that this need not be true for polytopes in general. Here we give topological Tverberg theorems for products of simplices. Each of these improves upon the $(r-1)(d+1)$-dimensional threshold, even while imposing the structural condition that the “Tverberg faces” of the product are themselves the products of pairwise disjoint faces from each simplex factor. As before, refinement extends these results, and in particular their dimensional improvements, to products of arbitrary polytopes. As an example, if $d+1$ is a power of two then whenever $m\geq n\geq d+1$ and $m+n=3d+2$ we show that any continuous map $f\colon \Delta_m\times \Delta_n\rightarrow \mathbb{R}^d$ admits disjoint faces $\sigma_1,\sigma_2$ of $\Delta_m$ and $\tau_1,\tau_2$ of $\Delta_n$ such that $\cap_{i,j\in[2]} f(\sigma_i\times\tau_j)\neq \emptyset$. In the case of multilinear maps, our results imply partitions of grid-indexed point sets in $\mathbb{R}^d$ by specialized subsets with stronger intersection conclusions than given by Tverberg's original theorem. Lastly, we extend our results to van Kampen–Flores type theorems which impose dimensional restrictions on the faces of each product factor.

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression–reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

For a self-map (f\colon X\to X) of a finite connected CW complex, we introduce visible Nielsen numbers by pushing the Reidemeister trace to finite regular covers compatible with (f). Optimizing over such covers gives a visibility profile and three resolution thresholds, including the observer complexity (\operatorname{oc}(f)), the least cover degree that separates all essential Nielsen fixed-point classes. We identify finite-observer indistinguishability with twisted conjugacy in the profinite completion and determine the associated blind subgroup. We also realize nonzero Reidemeister traces that vanish for every finite observer, and show that observer complexity is unbounded on a fixed finite complex even when the induced maps on the fundamental group and homology, the Lefschetz number, and the Nielsen number are fixed. For toral endomorphisms we compute the complete visibility profile as an exact divisor staircase. For compatible maps of principal torus bundles, in the finite-Reidemeister regime, we identify the unique maximal resolving kernel and express (\operatorname{oc}(F)) in terms of the Nielsen number and the radical of the bundle characteristic class modulo the fibre map. This yields exact formulas for scalar dilations of integral Heisenberg nilmanifolds, together with observer entropy and a rational observer-complexity zeta function. Finally, we construct toral and Heisenberg systems with identical Nielsen and Lefschetz sequences, Nielsen zeta functions, differential eigenvalues, and topological entropy, but exponentially different observer-complexity sequences.

In higher category theory, lax colimits are often understood to be a more useful and powerful generalization of usual (homotopy) colimits, which can be recovered from the lax colimit by a suitable localization. However, lax colimits do not provide the correct notion of gluing from the geometric perspective. Indeed, they are incompatible with the notion of categorical dimension, the Gray tensor product, and other basic geometric operations. In this paper, we develop the theory of oriented colimits, which correct the defects of lax colimits, and agree with lax colimits in dimension less than or equal to one. In order to study oriented colimits, we introduce a version of the Grothendieck construction which is compatible with enrichment in the Gray tensor product. We prove that the Grothendieck construction induces an equivalence between cartesian fibrations and presheaves of $(\infty,\infty)$-categories, which is enriched in the Gray tensor product of $(\infty,\infty)$-categories. Oriented colimits simultaneously generalize the concept of lax colimits and the Gray tensor product, and differ from lax colimits in much the same way in which the Gray tensor product differs from the cartesian product. We demonstrate the necessity of oriented colimits by showing that various fundamental constructions in higher category theory fail to be lax colimits but are instances of oriented colimits. As applications, we classify higher-categorical principal bundles, represent higher dimensional adjunctions by bicartesian fibrations of $(\infty,\infty)$-categories, and obtain higher categorical versions of Quillen's Theorems A and B, which admit very natural formulations in our framework.

We determine the two cross-stage morphisms arising from adjacent Stiefel fiber sequences in the special linear tower. Over characteristic-zero fields, the first is the Wood morphism at even stages and vanishes at odd stages; the second is induced stably by the motivic Hopf element at even stages and is zero at odd stages. Complex realization consequently classifies oriented rank-$(n-2)$ vector bundles on the split quadric $Q_{2n-1}$ for $n\geq5$. We then apply these calculations to give exact corank-two splitting and efficient generation criteria for projective modules.

Let ${\mathbb{G}}$ be an oriented $\mathbb{P}$-divisible group over a noetherian $\mathbb{E}_\infty$-ring $R$, let $G$ be a finite group, and let $\mathcal{F}$ be a family of subgroups of $G$. We show that completion of $R({\mathbb{G}})_G$-modules at $\mathcal{F}$ agrees with algebraic completion at the ideal $I_{\mathbb{G}}(\mathcal{F})=\bigcap_{H\in\mathcal{F}}\mathrm{ker} (\pi_0R({\mathbb{G}})^{ G}\to \pi_0R({\mathbb{G}})^{ H}).$ For ${\mathbb{G}}=\mu_{\mathbb{P}^\infty}$ over $\mathrm{KU}$ this recovers the family completion theorem of Adams, Haeberly, Jackowski, and May, and for the trivial family the classical Atiyah-Segal completion theorem. The main input is a theory of support for points of the tempered character stack ${\mathbb{G}}\{{\mathbb{B}} G\}$, in the spirit of Segal's analysis of the prime spectrum of the complex representation ring: we show that the support of a point is a single conjugacy class of abelian subgroups of $G$, and that the points supported inside $\mathcal{F}$ are exactly the preimage of $V(I_{\mathbb{G}}(\mathcal{F}))$ under the affinization map. We also prove a version over locally noetherian geometric base stacks, in which the ideal is replaced by an open substack of ${\mathbb{G}}({\mathbb{B}} G)$, the analogue over such a base of $\mathrm{Spec}\,R({\mathbb{G}})^{ G}$, and which applies for instance to genuine equivariant topological modular forms.

We give a new proof that algebraic K-theory increases chromatic height by at most one, originally established by Clausen-Mathew-Naumann-Noel. Our argument proceeds by descent from the Lubin-Tate spectrum, for which the required vanishing follows from Hahn-Wilson's Quillen-Lichtenbaum result.

In this paper, we introduce the notion of sequential $m$-contiguity distance for finitely many simplicial maps as a higher analogue of contiguity distance. This invariant generalizes both higher contiguity distance and $m$-contiguity distance, and provides a combinatorial counterpart of sequential $m$-homotopic distance. We investigate its fundamental properties, including invariance under strong homotopy type, behaviour under compositions, categorical products, and barycentric subdivision. Moreover, we define sequential $m$-discrete topological complexity of simplicial complexes. As applications, we characterise this invariant (along with $m$-simplicial LS category) in terms of sequential $m$-contiguity distance and prove that they are invariants of strong homotopy type. Furthermore, we establish inequalities relating $m$-simplicial LS category and $m$-discrete sequential topological complexity, extending classical results from topological complexity theory to the simplicial and $m$-dimensional setting.

The pencil diffeological space is a plane whose smooth structure at the origin is detected only along lines through it. We show that the Zariski cotangent space at the origin is entirely determined by the derivatives along these lines. Moreover, the derivatives along different lines form precisely a $\sigma$-continuous family, a weak form of continuity. We also compute the external and right tangent spaces at the origin from this description. This suggests that the local smooth structure at a singular point of a diffeological space can give rise to unexpected topological notions.