Papers

The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.

We give a desingularization-free proof of Voevodsky's identification of bistable mod-$\ell$ motivic cohomology operations with the motivic Steenrod algebra over fields of characteristic zero. This supplies the direct argument anticipated by Voevodsky in his study of motivic Eilenberg–MacLane spaces. As an application, we study Thom's topological obstructions to desingularization, which arose in his work on the Steenrod realization problem. Without invoking desingularization, we prove that the primary and all higher Thom obstructions vanish on complex algebraic cycles. The proof uses the motivic bidegrees of the successive $k$-invariants in the Brown–Peterson tower to show that every lifting obstruction vanishes.

In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits $\varprojlim$ and $\varprojlim^1$: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown–Grossman proper fundamental group of any open contractible $3$-manifold other than $\mathbb{R}^3$ is uncountable and perfect.

We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite exponent result applies to $\mathrm{NK}_n(R[G])$ whenever $R$ is a regular, torsion-free, Noetherian commutative ring such that $R/p$ is regular for every prime $p$ that divides $|G|$. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group $\mathrm{NK}_n(X)$ of an excellent Noetherian scheme $X$, with $X[1/p]$ regular, is annihilated by a finite power of $p$, provided $X$ admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for $R$ as above and $\alpha$ any automorphism of $G$, the Farrell Nil groups $\mathrm{NK}_n(R[G],\alpha)$ have finite exponent. As a consequence, for every virtually cyclic group $\Gamma$, the group $\mathrm{K}_n(\mathbb{Z}[\Gamma])$ is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.

We prove that all Morita classes $\mu_k \in H_{4k}(Aut(F_{2k+2});\mathbb{Q})$ are nonzero using a novel description of the Hopf algebra structure on Steinberg homology. We prove analogous results with twisted coefficients. Our results disprove a case of a finiteness conjecture of Kontsevich.

A large $p$-subgroup of a group $G$ is a self-centralizing $p$-subgroup $Q \le G$ whose normalizer controls the normalizers of all the non-trivial, central subgroups of $Q$. In 2016 Meierfrankenfeld, Stellmacher and Stroth produced a result describing the $p$-local structure of a finite group having a large $p$-subgroup (the main examples arising from finite groups of Lie type in defining characteristic $p$). This result is a major success within the wider framework of studying groups of local characteristic $p$. We attempt to produce a result analogous to that of Meierfrankenfeld, Stellmacher and Stroth, but for fusion systems and localities. Previous work of Ellen Henke and the author shows that reasonable generalizations can be formulated and solved equivalently either within the realm of fusion systems or in the world of localities. In particular, in the present paper we set the stage for our analysis, showing how working within a locality grants a clear advantage: it allows to follow the same lines of reasoning as for a group. We therefore produce analogous reduction results and case subdivision as those in the 2016 result of Meierfrankenfeld et al. and, proceeding according to the analogy, we deal with occurrences of certain natural orthogonal modules and with the first of the cases that are to be studied, the so-called symmetric case. The remaining cases will appear in future publications.

Restriction to the full cusp divisor of a modular curve defines a secondary invariant whose rational indeterminacy comes from a single global modular form. For every integral weakly holomorphic level-one modular form $h$ of weight divisible by four, we prove that the imported value $[h/2]$ vanishes at every nontrivial $\Gamma_0(N)$ level. The construction localizes coefficients before completion and rationalizes only after passing to homotopy groups. At odd prime level, we identify the full cusp spectrum as a product of two real Tate factors and construct a single holomorphic weight-two correction. This correction yields an actual integral global homotopy class and a rational global source class satisfying the equality required for joint annihilation. The equality transports to every odd composite level, while even levels follow by inverting two. At level three, the Mahowald–Rezk homotopy calculation leaves only the periodic $\nu$ family in stems $8K+3$. A cusp-residue homomorphism detects its secondary value, which has exact order two throughout the periodic family. As an application, the Bunke–Naumann secondary invariants of products in bidegrees $(8k+1,8k'+2)$ vanish after passage to every nontrivial level.

We construct canonical chiral $\mathcal N=(0,1)$ sigma models on Spin(7) manifolds. The Spin(7) structure fixes the rank-seven bundle of the left-moving fermions, whose quadratic index matches the tangent representation, so the internal anomalies cancel while the rank difference leaves the gravitational anomaly of one right-moving Majorana–Weyl fermion. A transverse section of the left-moving bundle has a 1D zero locus with induced String structure, and we compute its class in the first torsion group of topological modular forms. The class is nonzero precisely when the Euler characteristic of the target is odd. The standard Joyce action on the torus has no internal finite-group anomaly and can be gauged to define an exact chiral spin QFT. We show that the orbifold Euler characteristic is even throughout the diagonal Joyce family generated by coordinate reflections and half-shifts preserving the Cayley form. By contrast, a free antiholomorphic quotient of the Fermat sextic is a compact torsion-free Spin(7) target with Euler characteristic 1305 and realizes the nonzero class.

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical “straightening” construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato–Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

We construct a faithful functor from the category of cubespaces, arising in structured ergodic theory and nilspace theory, to the category of (condensed) simplicial sets. This functor translates the weak structure theorem of a nilspace to a Postnikov tower of the corresponding simplicial set. In particular, nilspaces are mapped to Kan complexes and the structure groups of the nilspace correspond to the simplicial homotopy groups of the Kan complex. The functor is constructed via pullback along a special cosimplicial cubical set, whose combinatorial properties may be of independent interest.

Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.

Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk–Ulam theorem for $O(k)$: every $B_k$-equivariant map from $O(k)$ to a natural representation of the hyperoctahedral group $B_k$ has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].

Let $M_{\ell,m}=\ell\,\mathbb{HP}^{2}\# m\,\overline{\mathbb{HP}^{2}}$ be the connected sum of $\ell$ copies of $\mathbb{HP}^{2}$ with $m$ copies of $\mathbb{HP}^{2}$ endowed with the opposite orientation. We characterise, in terms of Pontryagin classes, the real vector bundles over $M_{\ell,m}$ admitting a stable complex structure, and deduce that $M_{\ell,m}$ is stably almost complex if and only if $\ell-m$ is even. We then determine when an oriented real vector bundle of even rank over $M_{\ell,m}$ admits a complex structure. Consequently, $M_{\ell,m}$ admits an almost complex structure if and only if $\ell=2m+1$ and $m$ is odd. This corrects an assertion made by Sato and Suzuki in 1974, according to which $M_{\ell,m}$ is never almost complex. We also identify the unjustified step in their argument.

We provide a new presentation for a cylinder object inside of the category of co-globular $\infty$-groupoids. We prove that the cylinder construction that Lanari provides is isomorphic to the one we build in this paper.

Let $M=\widetilde H^*(MO(2);\mathbb F_2)$ and let $\mathcal A(1)$ be the subalgebra of the mod $2$ Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. In this work, we determine the quotient $\mathbb F_2\otimes_{\mathcal A(1)}M$ in every internal degree $h\geq0$, where internal degree $h$ corresponds to cohomological degree $h+2$. To overcome the limitations of finite experimental extrapolation, our proof is strictly certifying: in each degree, we exhibit a determinant-one minor of the binary hit matrix and a complementary family of linear dual functionals annihilating every hit column, mathematically guaranteeing the exact rank for all degrees. This rigorously establishes the monomial generating families and module decomposition proposed in Shih's 2012 report. Furthermore, we derive a closed binomial formula for every Steenrod square from the length-two symmetric-polynomial model, correct a boundary omission in the $\mathcal A(0)$ calculation, compute the exact Hilbert series of the $\mathcal A(1)$-cohit, and prove an intrinsic direct-sum decomposition of $M$ into two stable $\mathcal A(1)$-submodules.

We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group $\mathrm{SO}^\circ(n,1)$. For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.

Class disentanglement–the separation of a representation's class-conditional point clouds along depth and over training–is measured by descriptive curves: the sentence such a study wants to write, layer l+1 is more disentangled than layer l, is an eyeball judgement with no null. We supply the inferential layer for a topological measurement of class overlap, the Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, from one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test–exact permutation tests in both directions, a guarded separation certificate the invariant requires, and a paired sign-flip test for comparative claims. Building that test taught a lesson outliving this invariant: its statistic must be scale-free. On the raw profile mass, which has units of feature length, 12,375 paired tests return 5,633 significant steps of which every one at the first epoch points the wrong way, certifying feature-norm dynamics as disentanglement; the dimensionless statistic returns 2,707, with 2,026 decreases. Across 111 networks and 52,650 measurements, disentanglement is depth-graded and early, and interaction quotients rank class pairs by confusability (rho=0.83), on par with cheap separability statistics. In a 96-model factorial, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating. Only a k-fold statistic can pose the structural question: the joint entanglement of a class triple sits below its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, at median ratios far below a measured null floor, in vision encoders and frozen language models–a regularity, not a law. The unnormalized mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe.

We prove Morita invariance of the Campbell–Lind–Malkiewich–Ponto–Zakharevich restriction-system trace after passage to perfect modules. It therefore defines an intrinsic integral restriction trace for an exact endofunctor of a small idempotent-complete stable $\infty$-category. On $\pi_0$, the $m$-th ghost is the laced trace of the $m$-fold iterate, compatibly with Frobenius. For lattice-graded algebras, the ghost targets have twisted cocenters in degree zero; over the open parameter torus, this applies to the cyclic bimodule of Dinkins–Karpov–Krylov. For finite monomial endomorphisms of toric varieties, we construct motivic restriction classes whose ghosts are sums over cones fixed by the iterates. In one example, two classes have the same first ghost, while their second ghosts differ after rational Betti realization.

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\C[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.

We construct a map between simply connected rational spaces of finite rational type which admits a homotopy retraction and satisfies $msecat(p)=2<secat(p)=3$. The target is the rationalization of a simply connected twelve-dimensional CW complex with eleven cells, and is rationally hyperbolic. This disproves Carrasquel–Vera's conjecture for retractive maps.

As AI-code assistant tools become widespread, automatic assessment of the correctness of generated code becomes a significant challenge. Code LLMs are prone to hallucinations, which may lead to code that does not solve the required problem, or even to code with severe security vulnerabilities. In this paper, we introduce CodeTD – the first approach to pre-execution assessment of code correctness based on topological data analysis (TDA) of Code LLMs' attention maps. Our method quantifies prompt-generation mismatch using topological patterns of attention maps. We carry out experiments with common benchmarks (HumanEval, MBPP, BigCodeBench, MultiPL-E), 5 programming languages and 10 Code LLMs of size up to 34B parameters. The experimental results show that the proposed method outperforms recent baselines. Moreover, CodeTD is transferable between coding benchmarks.

We study the existence of maps of nonzero Brouwer degree between two orientable, equal-dimensional generalized Dold manifolds (GDMs) fibered by complex partial flag manifolds over real projective spaces. The GDMs considered here are obtained as orbit spaces of the diagonal involution on the product of a sphere and a complex partial flag manifold, acting antipodally on the sphere and by complex conjugation on the flag manifold. We show that no map of nonzero degree exists between two distinct GDMs when their base real projective spaces are different, except in one exceptional case. Furthermore, when the base real projective spaces coincide but fibers are distinct, we prove the nonexistence of maps of nonzero degree in most cases, including those in which at least one of the fibers is not a Grassmannian. As an application of our study, we establish cohomological rigidity for the class of products of a sphere with a complex or quaternionic partial flag manifold.

Building on the dagger Bergner and dagger Joyal models constructed in Part I, we develop simplicial-space models for dagger $(\infty,1)$-categories. We first construct a homotopically constant resolution model and a model structure for unitarily complete dagger Segal spaces. A change-of-index adjunction compares the two presentations of dagger simplicial spaces. We then prove a dagger analogue of the Joyal–Tierney Quillen equivalence. Together with Part I, these results give a chain of Quillen equivalences between the dagger Bergner, dagger Joyal, and dagger Rezk models.

Using the strong discrete Morse theory developed by Fernández, we define the complex $\mathcal{SM}(K)$ of strong discrete Morse matchings on a simplicial complex $K$, as well as the pure subcomplex $\mathcal{SM}_{pure}(K)$ generated by the facets in $\mathcal{SM}(K)$ of maximal dimension. For most complexes $K$ these objects are proper subcomplexes of the complexes ${\mathfrak M}(K)$ and ${\mathfrak M}_{\textrm{pure}}(K)$ defined by Chari–Joswig using all Morse matchings on $K$. The homotopy types of the latter are not well-understood in general, but they are known when $K$ is the path $P_n$ with $n$ edges, the cycle $C_n$ with $n$ edges, the star $S_n$ with $n$ leaves, the $n$-simplex $\Delta^n$ ($n\le 3$), and the boundary $\partial\Delta^n$ ($n\le 3$). in this paper we compute the homotopy types of $\mathcal{SM}_{pure}(C_n)$ and $\mathcal{SM}(K)$ for $K=P_n,S_n,\Delta^n,\partial\Delta^n$ for all $n$. We also compute the homology of $\mathcal{SM}(C_n)$ for $n\le 21$.

The Lusternik–Schnirelmann category of a closed symplectic manifold admits various estimates. Locally conformally symplectic (LCS) geometry is a generalization of symplectic geometry involving a closed one-form, called the Lee form. In this paper, we introduce Farber's Lusternik–Schnirelmann category for closed one-forms into LCS geometry by applying it to the Lee class. In particular, this provides a new topological approach to LCS structures of the second kind. We then obtain lower bounds for this invariant under LCS blow-ups.

Let $X$ be a $(q-1)$-connected rational space of finite type, where $q\ge2$, with homotopical nilpotency $nil_h(X)=n\ge1$. If $H^{>N}(X;Q)=0$ and $N\le q(n+3)-3$, we prove that $Hnil(X)=n$. More precisely, the sufficient bound is $N\le a_n(M_X)+2q-3$, where $a_n(M_X)$ is the first nonzero internal cohomology degree of $(M_X^+)^{n+1}$ in the minimal model. A family of examples proves that the constant $-3$ in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.