Papers

Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott–Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.

Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ is supercritical in the sense of continuum percolation, then the first homology group $H_1(M)$ of $M$ can be correctly inferred from $G_M(n;r)$ with high probability as $n \to \infty$. Specifically, one can obtain $H_1(M)$ by taking the cycle space of $G_M(n;r)$ and quotienting out all the cycles of metric diameter $O(r|\log r|)$ (or of graph diameter $O(|\log r|)$). Our method of estimating $H_1(M)$ exploits a coarse-topological fact about supercritical percolation, as opposed to usual methods, which examine the topology of neighborhoods of the point cloud. Whereas previous methods use combinatorial models which require $O(n \log n)$ edges, our method only requires $O(n)$ edges. We also show that, in all phases of the thermodynamic regime, if one instead takes the quotient by cycles of metric diameter $o(r|\log r|)$, with high probability, one will not recover $H_1(M)$. Thus $\Theta(r|\log r|)$ is the “right scale.” On the way, we show that an arbitrary compact $d$-dimensional Riemannian manifold has a first homological percolation threshold in the sense of Bobrowski and Skraba \cite{BS2020} which coincides with the continuum percolation threshold on $\R^d$, a result previously only known for the flat torus. This strongly suggests that our results are optimal, in the sense that $H_1(M)$ cannot be inferred from $G_M(n;r)$ in the subcritical thermodynamic regime. All results hold for homology with arbitrary coefficients.

Let $\mathcal{A}$ be a finite central real hyperplane arrangement and let $\mathcal{G}(\mathcal{A})$ be its tope graph. Koizumi recently proved that the crossing-graded magnitude homology of $\mathcal{G}(\mathcal{A})$ is torsion-free and is freely indexed by face flags. We refine his result by proving that a fixed crossing vector and terminal chamber determine a summand of rank at most one. Over $\mathbb{F}_2$, we use the canonical cohomology basis to determine the crossing-graded magnitude cohomology ring of $\mathcal{G}(\mathcal{A})$.

The topological Hedetniemi conjecture asserts that the topological index of the product of two \(\mathbb{Z}/2\)-complexes is equal to the minimum of their respective indices. Bui and Daneshpajouh studied a natural generalization of this conjecture to \(G\)-spaces and proved that this generalized conjecture fails whenever the group \(G\) is neither cyclic of prime-power order nor generalized quaternion. More precisely, for any such group \(G\), they constructed finite free \(G\)-simplicial complexes, each having topological index one, whose product has topological index zero. The purpose of this note is to demonstrate that this discrepancy can be made arbitrarily large. Indeed, for every integer \(n\geq 1\), we show that there exist two finite free \(G\)-simplicial complexes, each of topological index $n$, whose product has topological index zero, provided that \(G\) is neither cyclic of prime-power order nor generalized quaternion. In particular, the corresponding weak form of the generalized topological Hedetniemi conjecture fails in the strongest possible sense for those groups.

We study a nice class of Tambara functors, called multiplicatively cohomological Tambara functors. The first result is a proof of a conjecture of Balchin, Quigley, and Spitz showing that every multiplicatively cohomological Tambara functor is also additively cohomological. Building on this, we prove Nakayama's lemma for multiplicatively cohomological Tambara functors. As an application, we identify a condition under which finitely generated projective modules over local Tambara functors are always free. This applies, in particular, to the constant $C_{p^n}$-Tambara functor $\underline{\mathbb{Z}}_p$.

In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.

We provide proofs of several variants for spectra of the original Adams Conjecture. Our approach uses F-spaces (also known as Segal Gamma-spaces), resulting in establishing both unstable and stable homotopy equivalences between maps of spectra associated to the J-map and the Adams operations. We employ a rigid version of Artin-Mazur etale homotopy theory applied to commutative diagrams of simplicial schemes defined over the Witt vectors of fields of positive characteristic. Much of this text is devoting to developing a theory of X-fibrations and l-complete X-fibrations over the F-spaces we consider.

Point clouds arising in image collections, samples from Markov chain Monte Carlo, or states of a random walk, often have a simple underlying geometry which is obscured by noise, high ambient dimension, and the failure of Euclidean distance to reflect similarity. Methods such as UMAP and t-SNE condense such data into usable form, but rely on heuristic choices and provide no guarantee that the output reflects the topology of the input. We introduce a condensation method that comes with such a guarantee. Encoding the data as a positive $m\times n$ stochastic matrix $Q=(q_{ij})$, for instance the transition matrix of a random walk on the point cloud, we define a potential function $\psi(p)=\log \sum_{i} \exp(-KL(p,q_{i\bullet}))$ on the probability simplex $\Delta_n$, where $KL(p,q)$ is the Kullback-Leibler divergence, and prove that $\psi$ is $c$-convex in the sense of Optimal Transport Theory for the cost function $c(p,q)=KL(p,q)$. The associated transport map collapses noisy directions while provably preserving topology: the super-level sets of $\psi$ are homotopy equivalent to those of a $c$-conjugate function, whose image is a condensed, resampleable family of topological spaces which can be interpreted as a continuous analog of an alpha shape. We demonstrate the method by recovering the circle of camera angles from the COIL image dataset, where a standard PCA pipeline produces spurious homology, and the quotient $SO(3)/A_4$ from $45{,}000$ views of a tetrahedron in the SYMSOL pose-estimation benchmark.

We report our findings and the results of our experiment of autoformalizing the homotopy type theoretic computation of $\pi_3(S^2)$ in agda-unimath using Codex.

For $n$ odd, we enumerate rank $n$ complex topological vector bundles on $\mathbb CP^{n+2}$ with prescribed complex $K$-theory class. The answer exhibits a 24-fold periodicity and is closely related to the enumeration of corank $2$ stably trivial vector bundles.

We construct an Orlik–Solomon algebra for any locally geometric poset as a natural generalization of the one for geometric lattices. This algebra has several interesting features, including a combinatorial no-broken-circuit basis that we obtain through Gröbner basis theory. When the poset captures the intersection data of an arrangement of certain subgroups in a complex abelian Lie group, we infuse the Orlik–Solomon algebra with topological information to compute the cohomology of the arrangement complement. In particular, when the Lie group is compact, we present an explicit differential graded algebra whose cohomology is the rational cohomology of the arrangement complement. We further exhibit a family of abelian arrangements whose integer cohomology has torsion.

Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin–Lu–Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin–Lu–Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin–Lu–Yau curvature does not force higher-dimensional GLMY path homology to vanish.

We study the $\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\mathcal{H}^{\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable $\mathbb{P}^1$-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on $\mathbb{P}^1$. Over perfect fields, we then establish the $\mathbb{P}^1$-invariance of the higher unstable homotopy group sheaves $\pi_i^{\mathbb{P}^1}$ ($i \geq 1$) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable $\mathbb{P}^1$-connectivity theorem over fields. As a consequence, we obtain a homotopy $t$-structure on the stable $\infty$-category of $\mathbb{P}^1$-invariant Nisnevich spectra, and deduce the strict $\mathbb{P}^1$-invariance of stable $\mathbb{P}^1$-homotopy group sheaves.

We give three explicit constructions of the continuous Mal'cev Qp-completion of a topologically finitely generated pro-p group, using Tannakian formalism, Hopf algebras and p-adic analytic groups. We study properties of the continuous Mal'cev Qp-completion via these explicit constructions.

We study boundary-weighted barycenter spaces, in which an interior support point has cost (or weight) two and a boundary support point has cost one. These spaces are finite-dimensional topological models for the concentration patterns produced by noncompact boundary Euler–Lagrange functionals: an interior bubble carries twice the quantized mass of a boundary bubble, and very negative sublevels are therefore modeled by boundary-weighted rather than ordinary barycenters. The paper gives a systematic algebraic-topological treatment of these spaces. We construct the mixed strata, the closed-stratum poset, and the triangular boundary-weighted colimit filtration; compute the Euler characteristic; and prove a homology decomposition in terms of ordinary barycenter spaces of \(\partial M\) and \(M/\partial M\). We then specialize the formulas to compact connected orientable surfaces with boundary and to hemispheres, obtaining explicit rational Betti polynomials and, for surfaces, the mod-two polynomials required by the mean-field application. Finally, we explain how these polynomials measure the topology at infinity in the resonant Neumann mean-field equation on a compact surface with boundary.

We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)

A quantitative equational theory $U$ reasons about terms that agree up to a numerical error. It presents a free algebra $T_UA$ over a metric space $A$ of generators, the terms of the syntax at the least distance the axioms derive, and that metric is its semantic content. We give a functorial invariant of it, the persistent magnitude homology of $T_UA$: a barcode where the module is tame, finite linear algebra where $T_UA$ is finite, Lipschitz in each degree. Magnitude homology is graded by length and knows nothing of persistence, its persistent refinement nothing of where its bars begin and end, yet the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)\delta$ in degree $n$ under a perturbation of size $\delta$, and a computed perturbation shows that the factor cannot be dropped. An inclusion of theories induces a morphism of the presenting monads and, where the induced map is bijective and shortens no distance by more than $\delta$, a comparison of barcodes under the same bound, so a barcode movement measures the metric-semantic strength of the added axioms. Four examples are computed, one in every degree.

Anabelian geometry as an attempt to describe geometry in terms of étale topological data has addressed so far mainly categories of varieties over a field. In this paper we work over a normal base scheme $S$ of finite type over a sub-$p$-adic field and show that families of hyperbolic curves over $S$ are anabelian among smooth $S$-schemes with respect to dominant morphisms.

Let $G$ be a complex reductive group and $C_m(G)_1$ the identity component of the space of $m$-tuples of commuting elements in $G$. We prove that for every $m\geq 2$, the integral singular cohomology of $C_m(G)_1$ has torsion precisely at those primes which divide the order of the Weyl group of $G$. We deduce the analogous statement for compact Lie groups, settling a conjecture of Kishimoto and Takeda, and prove the corresponding result for compact Lie algebras. We prove these results using Smith theory for the conjugation action by semisimple elements of prime power order.

We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $\sigma$ via $\sigma$-colorings of triangulations, and we generalize Pachner's theorem in this context.

The Viterbo isomorphism relates the symplectic cohomology of a cotangent bundle to the homology of the free loop space of its base. We lift this to a relation of modules over (1) the complex bordism spectrum MU and (2) the sphere spectrum $\mathbb{S}$. In particular, by a result of Porcelli and the present author [BP26], it is not the case that, in general, the $\mathbb{S}$-level statement recovers the MU-level statement after base-change – even in the case the base is spin.

Let $\mathsf{P}$ be a finite graded poset and $\Delta_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\mathsf{P}$. We study the window convolution $C_a=\operatorname{Lan}_{q_1}\circ q_2^\ast$ on $\mathrm{Shv}(\mathsf{P};k)$, a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution $\mathbb{C}_a$ to compose as a flow, $\mathbb{C}_a\mathbb{C}_b\simeq\mathbb{C}_{a+b}$; where the meet assignment $\Phi$ is total it is a functor and carries a comparison map. Finality is sufficient, and necessary at every minimal apex and wherever the finality defect of $\Phi$ is essential; where $\Phi$ is total at a minimal apex with unit windows, it is the failure of a length-two interval above the apex to have a single interior element. The flow fails at every branching length-two interval with minimal bottom element, and with it on the face poset of every finite regular cell complex of dimension $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{C}_a$ gives a canonical extended interleaving pseudometric on $\operatorname{D^{b}}(\mathrm{Shv}(\mathsf{P};k))$; in the saturation cases computed here it takes no finite value above the length of $\mathsf{P}$, and is finite if and only if the derived colimits agree.

We give a short, conceptual account of Dwyer–Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer–Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.

We explain the relation between the relative topological Hochschild homology $\THH(R/\Sphere_{W(k)}[x_0,\ldots,x_n])$ and the Nygaard completed Frobenius twisted relative prismatic cohomology $\widehat{\Prism}^{(1)}_{R/W(k)[x_0,\ldots,x_n]^\wedge}$, where $W(k)[x_0,x_1,\ldots,x_n]\rightarrow R$ is relatively quasiregular semiperfectoid. As an application, for $R=\Z_p[x]/(px)$, we compute $\pi_*\THH(R)^\wedge_p$ by descent along $\THH(R)^\wedge_p\rightarrow \THH(R/\Sphere_p[z,x])$, where $R=\Z_p[x]/(px)$ is regarded as an $\Einfty$-$\Sphere_p[z,x]$-algebra through $\Sphere_p[z,x]\xrightarrow{z\mapsto p,x\mapsto x}\Z_p[x]/(px)$.

Following \cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots. A computer-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $\omega=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$. We relate that count to the rational signed weight enumerator $F_\omega(t)=\sum_{\eta\preceq\omega}(-1)^{\elln(\eta)}t^{|\eta|}$. For the counterexample, $F_\omega(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded. We prove a number of results and formulate a complete conjecture describing the above homology.

Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.

The main purpose of this paper is to provide a simple proof of the isometry theorem for one-parameter persistence modules, in which a matrix representing one morphism of an interleaving is reduced and the pivots determine a matching between barcodes. This approach applies to persistence modules of finite type indexed by the reals, and the more general statement for q-tame modules can then be deduced from it using approximations of q-tame modules. The similarity between this use of matrix reduction and that in the persistent homology algorithm motivates some further development of matrix computations for persistence modules, formalized by a category of barcodes in which the morphisms are equivalence classes of matrices. A method for computing induced maps on persistent homology is provided using matrix operations that fit naturally into persistent homology computations, making functorial persistent homology barcodes computable. A method is also given for computing persistent homology barcodes and morphisms when chains do not necessarily have infinite death times.

For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

We study the set of degrees of $\mathbb{Z}/m$-equivariant maps from spheres to complex Stiefel manifolds, motivated by the work of Astey–Gitler–Micha–Pastor. Under a suitable arithmetic condition, this set is determined using results of James, Atiyah–Todd, and Adams–Walker. Our approach is homotopy-theoretic.