Papers

A pair of topological spaces $X$ and $Y$ is a Fourier–Mukai (FM) pair if there is an equivalence of $\infty$-categories of derived sheaves of abelian groups \[\operatorname{Shv}(X;\mathrm D(\mathbb Z))\simeq \operatorname{Shv}(Y;\mathrm D(\mathbb Z)).\] An FM pair $(X,Y)$ is non-trivial if $X$ and $Y$ are not homeomorphic. We consider the question of whether non-trivial FM pairs $(X,Y)$ exist for which $X$ and $Y$ are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if $X$ is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of Tate–Hochschild cohomology (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold $X$, we relate the Tate–Hochschild cohomology of $\operatorname{Shv}(X;\mathrm D(\mathbb Z))$ to the homology at infinity of $X$ (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory $\mathcal D_{\mathrm{Gor}}\subseteq\mathcal D$ called the Gorenstein locus, defined for every dualizable stable category $\mathcal D$. For a triangularizable space $X$ and a field $k$, we show that the Gorenstein locus of $\operatorname{Shv}(X;\mathrm D(k))$ is equal to the full subcategory of sheaves supported on the $k$-homology manifold locus of $X$.

We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.

We prove comparison results about $\infty$-categories of derived commutative rings with extra algebraic structure. For example, we show that several notions of derived $\delta$-rings agree and obtain similar results for derived $\lambda$-rings, derived binomial rings, and derived Boolean rings. Our comparison results apply more generally to modules over a specific kind of monad on derived rings that we call derived plethory.

We compute the first homology of the unordered configuration space of $n$ open unit-squares in the $w$ by $h$ rectangle for all positive integers $n$, $w$, and $h$. Letting $w$ and $h$ vary, these square configuration spaces form a bi-filtration of the unordered configuration space of $n$ points in the plane, so their fundamental groups approximate the $n$-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if $w,h\ge 3$ and $n\le wh-3$. In the $h=2$ case, we show that the first homology groups of these square configuration spaces are free of rank $n-1$ for $n\le 2w-2$, whereas they are free of rank $w-1$ for $n=2w-1$.

We revisit the relation between the Kontsevich commutative graph complex and the homology of $\GL(n,\Z)$ found by Brown–Chan–Galatius–Payne using the sharbly complex of Ash–Miller–Patzt.

Let $A$ and $X$ be connected based CW complexes with $H_1(A;\mathbb Q)\neq 0$, and let $m\geq 1$ be the least degree with $\widetilde H_m(X;\mathbb Q)\neq0$. We prove that the Bousfield–Kan rational completion map of $A\vee X$ induces a map on $H_{2m}(-;\mathbb Q)$ whose cokernel has dimension at least $2^{\aleph_0}$. In particular, $A\vee X$ is $\mathbb Q$-bad.

Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.

We show that integers act invertibly on a derived scheme $S$ if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

In this paper, we study the rational realization of the gap-two product $K(\Q,n)\times K(\Q,n+2)$ as the classifying space $\B(X)$ for a simply-connected $\pi$-finite space $X$ with rational homology of finite type. We prove that $K(\Q,n)\times K(\Q,n+2)$ can be realized as $\B(X)$ up to rational homotopy equivalence if and only if $n=3,4$ or $4m+2$ for $m\geq 1$. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such $X$ is unique for $n=3$ or $4m+2$ for $m\geq 1$; there exist infinitely many rational homotopy types of such $X$ for $n=4$.

We study several fundamental properties of the category $\mathbf{Kur}$ introduced in \cite{Kim1}. By defining fibrations and weak equivalences, we show that morphisms with manifold targets admit homotopy fiber products; this is then applied to obtain a notion of generalized intersections of two submanifolds. A definition of the Weinstein category in the context of Kuranishi spaces is also presented. Finally, we establish that $\mathbf{Kur}$ forms a category of fibrant objects in the sense of Brown \cite{Brown}, and discuss its homotopy category $\operatorname{Ho}(\mathbf{Kur})$.

For $n\ge1$, we study representations of a simplicial group $G$ in the full $\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\mathbf{B}G$ and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map $f:G\to H$ is an equivalence, then $\pi_0(f)$ is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every $n\ge2$ and prime $q$ invertible in $k$, the nontrivial source $K(\mathbb Z/q,n-1)$ has the same $n$-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus $n$-representations depend only on the $n$-truncation of $\mathbf{B}G$, but need not determine that truncation, even together with vertex evaluation.

We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $\mu_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $\mu_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.

Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of ${\mathcal E}(M(\ell))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton–Kreck for closed, oriented $4$-manifolds.

Let $f:(\C^n,0)\to(\C,0)$ be a holomorphic germ with an isolated critical point at the origin, and let $\varphi_f$ be the negative gradient flow of $\re f$ with respect to a Riemannian metric. We show that the origin is an isolated invariant set of $\varphi_f$ and that its Conley index is the suspension of the Milnor fiber, \[ h(\varphi_f,\{0\})\;\simeq\;\Sigma F_f\;\simeq\;\bigvee_{\mu(f)}S^n , \] so that $\mu(f)=\rank \CH_n(\varphi_f,\{0\};\Z)$: the Milnor number is the rank of the homological Conley index of the flow in degree $n$. The proof uses continuation to a holomorphic morsification of $f$. Since $\im f$ is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the $\mu(f)$ nondegenerate rest points that result, and the sum property of the Conley index applies. McCord's theorem then recovers the Poincaré–Hopf index of $-\nabla\re f$.

Let $\mathcal{G}(G,A,\alpha)$ be a discrete 2-group, given as the {categorical} extension of an ordinary group $G$ by an abelian group $A$. We define principal $\mathcal{G}$-bundles over a manifold in terms of Čech data over a good open cover. In the case of the manifold $S^2$, we calculate this bicategory explicitly, and identify its 2-skeleton. We compare with the holonomy bicategory $\mathrm{Fun}(*/\!\!/ \pi_{\le 2}(S^2), * /\!\!/ \mathcal{G})$, and provide a concrete equivalence between the two.

We disprove the classical version of the Borsuk-Ulam-flavored conjecture, due to Baum-Dąbrowski-Hajac, that free actions $X\times \mathbb{G}\to X$ on non-empty compact Hausdorff spaces by non-trivial compact groups admit no equivariant maps $X*\mathbb{G}\to X$. Specifically, among non-trivial compact abelian $\mathbb{G}$, the conjecture fails precisely when $\mathbb{G}$ is torsion-free profinite.

Let $H_n$ be the one-dimensional Honda formal group of height $n$ over $\mathbf F_{p^n}$ and let \[ R_n=W(\mathbf F_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p). \] We prove, for every height $n$ and every prime $p$, that the prime ideals of $A_n$ stable under an open subgroup of the Morava stabilizer group are exactly the height ideals $(u_1,\ldots,u_j)$. We also prove that a stable prime of $R_n$ avoiding $p$ is zero. The resulting radical-ideal classification implies the Hovey-Strickland classification of thick tensor ideals in the category of dualizable $K(n)$-local spectra via the forward implication of Barthel-Heard-Naumann. The special-fiber argument is local and geometric. After cutting an invariant prime by a one-parameter curve, we construct from the Cartier structure equation a smooth formal quotient $\mathcal{Q}$ and a distinguished subgroup $\mathcal{H}$. The generic fiber of $\mathcal{H}$ is identified with the deformation space of connected-étale extensions. A Cartier obstruction map from the full Honda endomorphism order is compared with evaluation on Tate vectors through completed universal covers. Fargues-Fontaine vector bundles give a period-detection statement. Chai's rigidity theorem then promotes detection by homomorphisms to formal Zariski density after a renormalization of the valuation. A uniform fixed-jet argument transfers this density to Morava-stabilizer orbits. The generic-fiber assertion is proved separately from the Gross-Hopkins period map.

Farrell and Jones showed that negatively curved manifolds in dimension $\geq 5$ are topologically rigid in the sense that homotopy equivalence implies homeomorphism. In the smooth category, the result does not hold even for hyperbolic manifolds, and there are negatively curved manifolds $N$ homotopy equivalent to a given hyperbolic manifold $M$ but not diffeomorphic. For complex hyperbolic manifolds, the analogous construction is demonstrated only in dimensions of the form $8n+2$. In this paper, we prove this result in many other dimensions. The main approach involves the use of $EO$-theory of complex projective spaces to construct suitable examples of exotic spheres. These theories may be viewed as a version of real $K$-theory, and are defined at each prime $p$. The computation also yields positive results about the existence of free smooth $S^1$ and $S^3$ actions on exotic spheres which do not bound parallelizable manifolds.

In this note, for every topological colored nonsymmetric operad with CW spaces in each arity we construct a homotopy isomorphism with a CW homotopy operad (i.e. all higher homotopies are cellular with respect to the product structure and CW structure on cubes) not changing its objects. By using the two-sided bar construction, we rectify this to a homotopy isomorphism with a strict CW operad. This applies more generally to topological algebras over colored operads in sets. In the case of diagrams of finite CW complexes over finite posets, the resulting diagram consists of finite CW spaces enhancing the Quillen adjunction between simplicial sets and topological spaces. As a byproduct, we produce an embedding of topological homotopy algebras over operads into the presheaf category over the branch resolution for the category of $P$-trees generalizing the known case of topological operads.

We present a scalable method for computing persistence pairing graphs from large cubical filtrations and use it to test whether graph organization among persistent classes contains information beyond persistence intervals alone. The method cancels equal-value pairs before global reduction, constructs the compressed complex in streaming form, and recovers graph transitions from one persistence reduction through projected relations and triangular event solves. We apply the method to 119 signed distance volumes from the DRP-372 porous media collection for permeability prediction. Each 256-cubed volume induces more than 135 million cubical cells before contraction, whereas the median compressed complex contains 37,471 generators. In validation grouped by material family, adding persistence pairing graph descriptors improves prediction over persistence summaries under both Ridge regression and partial least squares regression. In the stricter experiment that holds out one source project at a time, Ridge retains a modest improvement over persistence alone, although additional benefit beyond geometry and persistence is not consistent across projects. Two null models show that the observed persistence and graph coupling and edge target organization are strongly nonrandom, but neither uniquely explains the predictive improvement; the useful signal appears to reside mainly in coarser graph and event organization. Spatial analyses likewise distinguish persistence magnitude from graph prominence and reveal nonlocal organization among algebraically related persistence events. These results show that persistence pairing graphs provide structurally informative summaries beyond the barcode while exposing domain dependence and representative sensitivity that motivate more invariant relational descriptors.

$\mathbb{Z}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}/p\mathbb{Z})$ class. In this paper we study the $\mathbb{Z}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the $\mathbb{Z}_p$-Thurston norm, which generalises the $\mathbb{Z}_2$-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of $\theta$. We define canonical $\mathbb{Z}_p$-folded surfaces, which connect the theory of $\mathbb{Z}_p$-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the $\mathbb{Z}_p$-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty $\mathbb{Z}_3$-folded surface embedded in $L(3n,1)$ is $2-n$ for $n\in\mathbb{Z}^{>0}$, determining the $\mathbb{Z}_3$-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.

We study the conjectural characteristic class analogue of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet-Schürmann-Yokura which expresses the Goresky-MacPherson homology $L$-classes in terms of suitable Hodge-theoretic $L$-classes, for quotient spaces $X/G$, where $G$ is a finite group and $X$ is a pure-dimensional complex projective variety. Assuming that an equivariant $K$-theoretical version of the conjecture holds for $X$, we show that the characteristic class conjecture holds for $X/G$. Without assuming the equivariant $K$-theoretical version, we show that the characteristic class conjecture holds for $X/G$, provided that it holds for all fixed point sets $X^g$ and that a suitable normally nonsingular inclusion assumption is satisfied. Our approach is to work in equivariant analytic $K$-homology and equivariant algebraic $G$-theory and to identify the corresponding localized classes. On the analytic side, we identify the Banagl-Zagier equivariant $L$-classes with localized Chern character of the equivariant $K$-homology class of the signature operator defined by Banagl-Leichtnam-Piazza. On the algebraic side, we compute the localization of the equivariant motivic Hodge-Chern class transformation of the intersection Hodge module.

We introduce three homology theories for hypergraphs, namely $\Gamma$-homology, $\Box$-homology, and $\times$-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that $\Box$-homology is invariant under this homotopy, whereas $\Gamma$-homology and $\times$-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by $\Gamma$-homology that further differentiates it from $\Box$-homology.

Given a topological space $X$ and a family of subgroups $\mathcal{F}$ of $\pi_1(X)$, the $\mathcal{F}$-category of $X$ is given as one less than the cardinality of the smallest cover of $X$ by open subsets whose fundamental groups lie in $\mathcal{F}$. In this article we study $\mathcal{F}$-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining $\mathcal{F}$-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of $\mathcal{F}$-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.

We construct cochain models for Serre fibrations over path-connected bases with coefficients in an arbitrary field. Our construction refines Brown's twisted tensor product to produce a differential graded module model over the singular cochain algebra of the base. If the homology of the fibre is finite dimensional in each degree and on which the action of the fundamental group of the base is degreewise nilpotent, then this model is a semifree resolution generated by the cohomology of the fibre. This extends the classical construction for simply connected bases. As applications, we compare the resulting model with arbitrary semifree resolutions, showing that every such resolution recovers the cohomology of the fibre after base change to the basepoint, and obtain an explicit bound on the page at which an associated spectral sequence collapses.

We construct a polytopal simplicial sphere that admits a mod 2 characteristic map but no mod 3 characteristic map, and hence no integral one. This shows that the Buchstaber number of a polytopal sphere can be strictly smaller than its real Buchstaber number, and gives a negative answer to the toric lifting problem.

We prove that the set of real $\mathrm{SL}_n$-opers forms a discrete subset of the moduli space of irreducible flat connections.

We completely determine the possible parities of the Euler characteristic of closed $Pin^{\pm}$-manifolds of a given dimension. As an application we partially calculate the controlled cut-and-paste groups $SKK_{2k+1}^{Pin^{\pm}}$, which were previously undetermined.

For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}_1$-ring spectrum $R_G$. When $G=C_{p^n}$ is a cyclic $p$-group, we provide an explicit construction of the $\mathbb{E}_1$-ring spectrum $R_{C_{p^n}}$. The proof uses a reconstruction theorem for genuine $C_{p^n}$-spectra, which might be of independent interest. As an application, we compute the $\mathbb{Z}$-graded $C_{p^n}$-equivariant mod $p$ Steenrod algebra.

We prove that the categories of strictly unital (resp.\ non unital) A$_{\infty}$-algebras and A$_{\infty}$-categories, with strictly unital (resp.\ non unital) A$_{\infty}$-morphisms and A$_{\infty}$-functors have no coequalizers.