Papers

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\mathbb{Z}_k$-smooth manifold representative, then $\Sigma$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}/k\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or $\mathbb{Z}_k$-smooth submanifolds, even in homology classes with smooth or $\mathbb{Z}_k$-smooth representatives.

The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the $(\infty, 1)$-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be $T_1$-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.

This paper addresses the computation of persistent homology for filtrations of cellular cosheaves over a fixed cell complex, where the algebraic data vary along the filtration while the underlying topological structure remains unchanged. We extend methods from discrete Morse theory to this setting, establishing conditions under which chain complexes can be reduced while preserving the associated persistence module. Building on these results, we introduce "FilteredScythe", an algorithm that identifies Morse matchings compatible with all stages of the filtration and performs reductions using Schur complements, based on the exist "Scythe" algorithm. We prove the correctness of these reductions and establish bounds on the algorithm's time and space complexity.

We show that every vineyard can be decomposed into vines, that is, every path in the space of persistence diagrams $\mathcal{D}$ can be decomposed into a sequence of paths in $\mathbb R^2$. Moreover, we characterize which spaces $X$ have the property that every continuous map $X\to \mathcal{D}$ has such a decomposition. Our results hold not just for $\mathcal{D}$, but also for many other spaces of unordered tuples.

We determine the $RO(C_2)$-graded $k\mathbb R$-Hurewicz image, where $k \mathbb R$ denotes the connective Atiyah's Real $K$-theory spectrum. As a consequence, we obtain four infinite families in the $RO(C_2)$-graded equivariant stable stems that are $C_2$-equivariant refinements of the classical $\mu$-families. Using a $C_2$-equivariant $v_{1,0}$-self map, we explicitly construct an equivariant Adams periodicity operator $P_{C_2}$ that captures these $v_1$-periodicity phenomena in the $C_2$-stable stems.

This is my written Laudatio for Graeme Segal on receiving the Chern Medal at the 2026 International Congress of Mathematicians. I make some general remarks about Segal's qualities as a mathematician. Then I discuss the origins of his work on classifying spaces of categories and Gamma-spaces. In the last section I describe his axiom system for Wick-rotated quantum field theory and some of his subsequent work in the area. Of course, this is necessarily only a small slice of Segal's total research output.

Symmetries of a quantum field theory or a quantum lattice model are expected to be described not by a group but by a homotopy type, whose homotopy groups are the higher symmetry groups and whose Postnikov invariants encode 't Hooft anomalies. We propose a homotopy-theoretic definition of 't Hooft anomalies based on the structure of localized symmetries in the operator-algebraic approach. The input is a crossed precosheaf: a collection of groups attached to regions of space, equipped with pairings recording that the commutator of two transformations localized in two regions is localized in their intersection. To a crossed precosheaf and a finite family of regions we attach a pointed connected homotopy type, the Cech-Loday classifying space. As the family of regions ranges over finite covers of space, the classifying spaces form a pro-object in the homotopy category. There is a canonical map from the ordinary classifying space of the symmetry group to this pro-type. Composing this map with a group action gives an anomaly whose nontriviality obstructs gauging. In favorable cases, the Cech-Loday space is essentially independent of the cover. In such cases, the homotopy fiber of the canonical map is a candidate for the classifying space of the higher group of symmetries. We give examples of crossed precosheaves arising from gauge transformations in continuum field theory and from quantum cellular automata and finite-depth circuits. For quantum spin systems in one and two dimensions, our construction recovers the Else-Nayak index and its generalizations to two dimensions.

Persistent homology summarizes the topological evolution of a graph as persistence diagrams (PDs). Generating graphs whose topology matches a target descriptor is increasingly important, yet existing topology-aware generative models rely on soft regularization, producing graphs whose PDs may only be approximately similar. We initiate a principled investigation of the generation problem: produce graphs that realize a target PD exactly. We propose two approaches guided by the implicit local and global constraints a target PD imposes. The local approach builds degree-based PD-equivalent graphs iteratively, and we prove that every output realizes the target PD and that every degree-based PD-equivalent graph is reachable. However, as shown thereafter, local generation necessarily trades off between sample diversity and reaching infeasible states. We therefore recast generation as a global constraint satisfaction problem with a complete encoding solvable by modern constraint programming solvers. This is applicable to any vertex-based permutation equivariant filtration scheme and is minimal under the degree filtration. Experiments on four standard graph generation benchmarks confirm that both methods realize the target PD exactly. Together, our methods provide a principled treatment of strict topology-compliant graph generation. Code is available at https://github.com/Aalto-QuML/ExactCompliance.

We propose a of definition conjugate stack for a class of suitable $\mathbb{E}_3$-$\mathrm{MU}$-algebras that includes the truncated Brown-Peterson spectra $\mathrm{BP}\langle n \rangle$ for all heights $n$. Then we calculate part of $\mathrm{BP}\langle n \rangle^{\mathrm{conj}}$ as a quotient stack in terms of Witt vectors. To perform this computation we study generalized Thom spectra and their relative Topological Hochschild Homology. We exhibit the Eilenberg-Mac Lane spectrum $\mathrm{H}\mathbb{F}_2$ as a $\mathbb{E}_3$-$\mathrm{MU}$-Thom spectrum, the $p$-local Eilenberg-Mac Lane spectra $\mathrm{H}\mathbb{Z}/p^n$ (except for $\mathrm{H}\mathbb{Z}/4$) and $\mathrm{H}\mathbb{Z}_{(p)}$ as $\mathbb{E}_3$-$\mathrm{MU}_{(p)}$-Thom spectra, and the connective complex K-theory spectrum $ku$ as $\mathbb{E}_1$-$\mathrm{MU}$-Thom spectra. We accompany these “go" results with “no-go" results that show these “go" results do not admit $\mathbb{E}_{\infty}$-refinements.

We show by explicit counterexample that the Becker–Gottlieb transfer cannot be made functorial. A different counterexample using rational homotopy theory was discovered concurrently by the third author and Robin Stoll.

We prove that the Becker-Gottlieb transfer is not functorial, even on rational cohomology. More precisely, for any even $n \ge 4$, we construct a fibration $f \colon X \to S^{n+1}$ with fiber $S^2 \vee S^n$, and an oriented $S^1$-bundle $g \colon Y \to X$ such that the transfer of $f \circ g$ is non-trivial on rational cohomology. Since the transfer of an oriented $S^1$-bundle vanishes, this implies that the transfer of $f \circ g$ and the composite of the transfers of $f$ and $g$ induce different maps on rational cohomology and are thus not homotopic. The proof that the transfer of $f \circ g$ is non-trivial uses the rational models for the Becker–Gottlieb and THH transfers obtained in previous work of the authors. In the appendix, we provide a second, more involved, proof that relies on constructing explicit $C_\infty$-algebra models for odd spherical fibrations (including circle bundles), which might be of independent interest. Concurrently, another counterexample (of very different flavor) was found by Georg Lehner, Cary Malkiewich, and the first author.

We consider the image of the comparison homomorphism from bounded cohomology to ordinary cohomology. Its dimension is called the bounded Betti number. In our earlier work we proved that the total bounded Betti number of a closed Riemannian $n$-manifold satisfying $\Ric\geq -(n-1)$ and $\Diam\leq D$ is bounded by a constant depending only on $n$ and $D$. The original proof used the amenability of the images of sufficiently small local fundamental groups. Since that work was written, Kapovitch and Wilking proved Gromov's generalized Margulis lemma under a lower Ricci curvature bound. Their theorem gives the stronger conclusion that these local groups are uniformly virtually nilpotent. We explain how their result supplies a direct modern input for the bounded Betti number theorem. We also discuss the topological meaning of the bounded part of ordinary cohomology and its relation to the $\ell^1$-seminorm on homology.

Let $\mathcal O$ be the Lubin–Tate deformation ring of a one-dimensional commutative formal group of finite height $n\geq1$ over a perfect field $k$ of characteristic $p$. We identify Rezk's Witt filtration $\{I_d\}$ with the divisibility filtration of integral trace pairings on subgroup algebras of fixed source lifts. This gives the geometric recursion \[ I_{d+1}=pI_d+J_{d,z}\qquad(d\geq0), \] where $J_{d,z}$ defines the targets of order-$p^d$ quotient isogenies from a $W(k)$-valued lift $z$ of the closed point. The corrected ideal is independent of $z$. We also compute the cokernel of the universal source trace pairing over $\mathcal O$: it is $\bigoplus_{j=1}^d(\mathcal O/p^j)^{r_j-r_{j-1}}$, where $r_j$ counts order-$p^j$ subgroups of $(\mathbf Q_p/\mathbf Z_p)^n$. The same duality extends the generic trace along two-step isogeny flags to an integral surjection, even when the flag-forgetting map is not flat. This computes composition in the subring of weighted trace operations. Applications include sharp projector denominators and congruence fibre products for full-torsion loci.

We propose the study of the class of groups that fix a point for every continuous action on any convex compact space. Oliver completed the corresponding program of classifying the finite groups that have this fixed-point property in finite dimensions. Turning to infinite groups, the Schauder–Tychonoff theorem establishes that the integers have the fixed-point property, but this has not been shown for any other infinite, finitely generated group. We establish a local transfer principle and use it to show, as a proof of concept, that $\mathbf{Z} \times (\mathbf{Z}/p\mathbf{Z})$ belongs to this class of groups.

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

The independence complex of the Kneser graph $\KG(n,k)$ is the simplicial complex of pairwise intersecting families of $n$-subsets of $[2n+k]$; equivalently, it is a Vietoris–Rips complex of the $n$-subsets of $[2n+k]$ under the symmetric-difference metric. Feng and Wang asked whether, for $n=3$, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every $k\ge0$, the only nonzero reduced Betti numbers using rationals of $\Ind(\KG(3,k))$ are \[ \beta_6=\binom{k+5}{7}+29\binom{k+6}{7} \qquad\text{and}\qquad \beta_9=\binom{k+6}{6}. \] The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer–Vietoris spectral sequence which reduces the problem to the six-element sets.

We identify the $(\infty,1)$-category presented by the dagger Bergner, dagger Joyal, and dagger Rezk models of Parts I and II with that of univalent flagged dagger $(\infty,1)$-categories. We first construct a realization–nerve adjunction between reversal simplicial anima and Hermitian flags. We prove that its nerve restricts to an equivalence between flagged dagger $(\infty,1)$-categories and dagger Segal anima, under which univalence corresponds to unitary completeness. We then construct the adjunctions for univalentization and Hermitian completion and identify univalentization with unitary completion.

We present an $\infty$-categorical refinement of classical $KK$-theory and $E$-theory for graded, $G$-equivariant for a locally compact second countable group $G$, Real $C^*$-algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$-theory spectra, and explicitly analyse the induced endomorphisms on $K_0$-groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$- and $E$-categories are not a Dwyer-Kan localization of graded $C^*$-algebras.

The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to $\C^N$, with prescribed finite nerves, or are unions of Fatou–Bieberbach domains whose nonempty finite intersections are again Fatou–Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some $\C^N$, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in $\C^5$. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.

We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold $\RP^2\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel $\pi_4(\RP^2\times S^2)\cong(\Z/2)^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton–Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to $\RP^2\times S^2$ and to $\RP^4$, and for the Hambleton–Hillman manifold with fundamental group $(\Z/2)^2$ we determine the normal $1$-type and show that self-equivalences act on $\pi_2$ through diagonal sign matrices. For $\RP^2\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.

In this paper, we apply persistent homology to study finite topological spaces by associating persistence diagrams with their elements through the furtherness function. This approach enables us to classify elements according to their topological evolution within the space. We further establish a bottleneck stability theorem showing that the bottleneck distance between the persistence diagrams associated with two elements is bounded above by their mutual furtherness.

In this paper, we discuss equivariant formality for spaces equipped with a primary cyclic group action. We study the behavior of equivariant formality with some topological constructions. The algebraic complexity needs to deal with injectivity. In the end, we discuss equivariant intrinsic formality.

This article takes steps to show that an equivariant version of the well known phenomena of Koszul Duality can be formulated in a manner not considered prior: via dg (co)algebra valued functors on the orbit category. As such techniques used in this article mix traditional and modern categorical methods in homotopy theory.

A topological group, defined in terms of “composable minimal geodesics” is constructed and shown to be homotopy equivalent to the pointed loop space of a path-connected, simply connected Riemannian manifold $M$ of positive convexity radius. This construction is analogous to John Milnor's free group model for the pointed loop space of a connected simplicial complex. Related models built from composable minimal geodesics are also obtained for the free loop space of $M$ and for the space of maps from any finite two-dimensional CW-complex into $M$.

I introduce a new class of cell complexes, called CL-complexes, that provide alternative topological realizations of finite posets. Each realization of a given poset has the same homotopy type as the order complex of that, so in some sense these spaces do not provide anything really new. Their value is that they enable short transparent proofs of standard results in poset homotopy. These include Quillen's Theorem A for finite posets, the Nerve Theorem for good covers of a topological space, and Sheehy's multicover generalization of the Nerve Theorem. Order complexes retain a special status: the order complex of a finite poset admits CL-complex structures both for the poset itself and for its opposite poset. The definition of CL-complexes is motivated by an elementary generalization of the Carrier Lemma of J. W. Walker, which in its original form constructs a canonical homotopy class of maps from a simplicial complex to a space covered by a suitable family of contractible sets. In the generalized lemma, the role played by simplicial complexes and the role played by spaces covered by contractible sets can both be fulfilled by CL-complexes. This sets up an equivalence of categories that tacitly underlies our approach.

The $\mathbb{A}^1$-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local $\mathbb{A}^1$-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as $\mathbb{A}^1$-Milnor numbers. Over fields of characteristic not $2$, $3$, or $5$, we show that every element of the Grothendieck-Witt ring of rank at most $7$ that is a local $\mathbb{A}^1$-Brouwer degree is also an $\mathbb{A}^1$-Milnor number. Unlike for local $\mathbb{A}^1$-Brouwer degrees, however, in general not all $\mathbb{A}^1$-Milnor numbers can be obtained from two-variable polynomials. Over an arbitrary field of characteristic not $2$, $3$, $5$, or $7$, we characterize the elements of the Grothendieck-Witt ring of rank at most $8$ that are $\mathbb{A}^1$-Milnor numbers of two-variable polynomials. Over subfields of $\mathbb{R}$, this implies that there are local $\mathbb{A}^1$-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the $\mathbb{A}^1$-Milnor number of a two-variable polynomial. We also prove that under a mild hypothesis one can reduce the number of variables of a polynomial while preserving the $\mathbb{A}^1$-Milnor number up to multiplication by a unit.

We define a $d$-cube in a graph $G$ to be monophobic in $G$ if, whenever it supports a $d$-face of a discrete singular $d+1$-cube, it must also support another face. In many cases, the presence of such cubes implies the nontriviality of discrete cubical homology and homotopy groups associated with $G$. We describe several large classes of examples and characterize the property completely when $d=2$.

In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras $k[\Gamma]$ equipped with the involution $g \mapsto g^{-1}$, we relate the involutive representation homology of $k[\Gamma]$ to the representation homology of the group $\Gamma$ with coefficients in $\mu_2$, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of $B\Gamma$; more generally, we compute the representation homology of $\Gamma$ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of $k[\Gamma]$ is concentrated in degree $0$ in characteristic $0$, for every $\Gamma$.

The expressivity of a neural network $\Phi$ describes the class of functions that a network can express. The term topological expressivity refers to a family of measures particularly suited to understand binary classification and is often formulated in terms of the Betti numbers of a decision boundary $\beta (\Phi^{-1}(0))$. Despite its popularity, this measure is not as useful for regression and other problems that do not involve a decision boundary. Instead, we propose to study topological expressivity in terms of the Betti numbers of the output of a neural network $\beta(\Phi(X))$. We formally prove that a ReLU network with architecture $(d,\sum\limits_{k=1}^{d } \beta_k(3k+2),\sum\limits_{k=1}^{d}\beta_k(k+1))$ can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected ($\beta_0=1$) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of $N$ spheres of dimension $d_1,...,d_N$ we achieve much better bounds $(\sum\limits_{n=1} d_n, 3\sum\limits_{n=1}\limits^N d_n + 2N, \sum\limits_{n=1}\limits^N d_n + N) $ for Betti numbers given by $\prod_n (1+x^{d_n})$. Finally we conjecture a much stronger tight bound which gives the architectural bound $(d,d+1,d+1)$ for a space homeomorphic to the sphere.

Let $f$, $g\colon A\to B$ be two A$_{\infty}$-morphisms between two A$_{\infty}$-algebras (linear over a commutative ring $\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\infty}$-algebra $B\otimes R$, where $R:=\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\times B$ in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A$_{\infty}$-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A$_{\infty}$-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A$_{\infty}$-algebras with (non strict) A$_{\infty}$-morphisms, and we explain why weakly equivalent A$_{\infty}$-morphisms need not be homotopic.