Papers

The weak, stable, and ordinary Lusternik–Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\geq 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\leq a<b\leq 2a$ and $c\geq b$, as well as every triple $(a,a,c)$ with $2\leq a\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.

In this work, we introduce a new clustering method, namely T-ARC (Topology-Aware Randomized Clustering), that corrects the geometric bias of K-means by embedding topological information directly into the optimization objective. Building on the assumption that the data admits an underlying hidden structure modeled via a latent graph, the idea is to uncover this information through the interplay between the standard K-means data-fidelity term and a graph-cut penalty, which discourages cluster assignments inconsistent with the connectivity structure of the data. To render this coupling tractable, the latent graph is modeled as a random realization from a Stochastic Block Model (SBM), whose scalar parameter is optimized within a Distributionally Robust Optimization (DRO) framework, yielding a closed-form proximal update. Both SBM and DRO are informed by a persistence-based similarity matrix derived from zero-dimensional persistent homology ($H_0$), which translates the multiscale connectivity structure of the data into a pairwise topological prior. The overall optimization proceeds via Block Coordinate Descent; convergence is established through a global Lyapunov functional: the deterministic blocks satisfy monotonic descent, while the stochastic graph update satisfies descent in expectation, so that the expected energy converges. Experiments on synthetic datasets with non-convex geometries and on random subsets of Fashion-MNIST show that T-ARC recovers latent topological structures where K-means fails, achieving the highest accuracy on curved and interleaved clusters while remaining competitive, and markedly more stable than K-means, on real data.

Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash–Miller–Patzt, uses the sharbly resolution. The other, by Brown–Chan–Galatius–Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.

Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented $BO\langle n+1\rangle$-manifolds $M$ of dimension $2n$ or $2n-1$ whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of $\pi_1$ on the homotopy of the fibre of $M\to P_n(M)$. We compute $\operatorname{Aut}_\bullet(S^1\times S^{m-1})\cong(\mathbb{Z}/2)^2$ for $m\ge4$, and $\operatorname{Aut}_\bullet(S^3\times S^3)\cong(\mathbb{Z}/12)^2\rtimes SL_2(\mathbb{Z})$ with the standard action modulo $12$. Further, $\operatorname{Aut}_\bullet(S^3\times S^4)\cong(\mathbb{Z}/2\oplus\mathbb{Z}/24)\rtimes(\mathbb{Z}/2)^2$ has order $192$, that $\operatorname{Aut}_\bullet(S^4\times S^4)\cong D_4\times\mathbb{Z}/2$ has order $16$, and that for connected sums of $r\ge2$ copies of $S^3\times S^3$ the kernel contains a subgroup $(\mathbb{Z}/12)^{2r}$ on which the permutations of the summands act non-trivially. For $S^p\times S^p$ the kernel is a sum of two copies of $\pi_{2p}(S^p)$ modulo Whitehead products with $\iota_p$, and the image is determined by an obstruction on the Whitehead square. In particular $\operatorname{Aut}_\bullet(S^5\times S^5)$ is isomorphic to the theta subgroup of $SL_2(\mathbb{Z})$ and $\operatorname{Aut}_\bullet(S^7\times S^7)$ is an extension of $SL_2(\mathbb{Z})$ by $(\mathbb{Z}/120)^2$.

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$, and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. When the isomorphism of completions is compatible with an identification of the abelianizations, the loci correspond exactly; in general, they correspond up to an isogeny. Applied to finite covers, this shows that the tropical bounds for the Bieri–Neumann–Strebel–Renz invariants are profinite invariants, and recovers the profinite invariance of the BNS invariant of Kähler groups due to Hughes, Llosa Isenrich, Py, Stover, and Vidussi. We also show that $\widehat{G}$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show, respectively, that the profinite completion of an arrangement group is not combinatorially determined, and that it does not determine the group.

We compute the Dolbeault cohomology of minimally stable polytopal LVM manifolds using a finite curvature complex. Let $B$ be the indispensable weight block. A canonical exact sequence identifies the kernel of the curvature map with $(\operatorname{coker}B)^\vee$ and its cokernel with $(\ker B)^\vee$. At full rank, the Dolbeault groups are direct sums of reduced cohomology groups of induced subcomplexes of the visible simplicial sphere, tensored with the exterior algebra on the complex dual of the deck lattice. One proof uses the curvature model and Stanley–Reisner Tor; a second gives an additive comparison through completed Laurent expansions and holomorphic descent. Full curvature rank is equivalent to Frolicher degeneration at $E_1$. Every manifold in this class fails the $\partial\bar\partial$ lemma. For polygons, the number of facets and the curvature rank determine the entire Hodge diamond. We construct a proper holomorphic family over a disk with fixed visible square whose curvature rank drops at the origin. The total dimension of Dolbeault cohomology increases there by sixteen. At full rank, holomorphic descent also computes tangent-sheaf cohomology, including resonant contributions in positive Cech degree. We determine the Kodaira–Spencer map of the normalized weight family and give a cohomological criterion for semiuniversality with smooth base. For a resonant example over the square, the global vector fields have a polynomial basis of nine elements and $h^1(\Theta)=19$. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.

The nerve theorem provides a combinatorial model for the homotopy type of a topological space from a suitable open cover. We extend this local-to-global approach by replacing the intersections of the cover with compatible categorical models. These local models are assembled through the Grothendieck construction, yielding a small category whose classifying space has the weak homotopy type of the original space. Under suitable hypotheses, the resulting category is finite and acyclic. We develop two versions of this construction. The first is indexed by the usual poset of nonempty finite intersections of the cover. The second uses the membership poset, inspired by the finite-space construction of Sancho de Salas, which retains only the membership patterns realized by points of the space and is defined for point-finite covers. The classical nerve theorem and its componentwise variant are recovered as particular cases. The construction is recursive: categorical models of local pieces, together with functors representing their inclusions, can be combined to produce categorical models of more complex spaces.

We introduce the tangloid algebra $\mathcal{T}_n$, a diagrammatic algebra arising from the extended tangloid category $\Ti$, and its subalgebra, the braidoid algebra $\mathcal{B}_n$. The construction begins with the unoriented tangloid category $UTC$, inspired by Turaev's theory of knotoids, which is obtained from the unoriented welded tangleoid category $UWTC$ by removing the welded relation while retaining the forbidden moves and imposing some additional relations. Within this framework, the braidoid category arises naturally as a subcategory of $UTC$. We then introduce the extended tangloid category $\Ti$ by adjoining a placeholder morphism $\emptyset$, which allows the generators to be extended to endomorphisms of a fixed $n$-box. The interaction of the placeholder morphism with the crossing morphisms gives rise to diagonal morphisms, $/$ and $\backslash$, which are used in the defining relations of $\Ti$. The corresponding extended braidoid category $\mathbf{Brd}_{\Ti}$ is realised as a subcategory of $\Ti$. For each $n\geq 0$, we then define the tangloid algebra $\mathcal{T}_n$ by linearising the endomorphism algebra $\operatorname{End}_{\Ti}(n)$, as a unital $\mathbb{C}$–algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra $\mathcal{B}_n$, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra $\delta\mathcal{T}_n$ by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic bilinear pairing on $\delta\mathcal{T}_n$ using diagrammatic reflection, composition, and closure. Our categorical and algebraic setting provides a theoretical framework for studying diagrammatic structures with intrinsic endpoints, namely knotoids, linkoids, and braidoids, and related diagram algebras.

We extend a six-functor formalism $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ to a lax symmetric monoidal functor of $(\infty,2)$-categories $\mathbf{Span}^2(C,E)^P_I\to\mathbf{Cat}$, where $P$ and $I$ are the classes of $D$-proper and $D$-étale morphisms, respectively. This proves a conjecture of Mann and generalizes a special case of a theorem of Cnossen, Lenz, and Linskens. To prove this result, we develop a theory of internal $\mathsf{E}$-monoidal categories and $\mathsf{E}$-operads, where $\mathsf{E}$ is a local class of morphisms in an $\infty$-topos. These notions generalize the internal symmetric monoidal categories and operads developed by Martini and Wolf.

We consider the moduli stack $\mathcal{H}_{g,d}$ parametrising maps $\theta\colon C\to L$ of degree $d$ between complex curves of genera $g$ and $0$. We provide a splitting of the stable rational cohomology of $\mathcal{H}_{g,d}$, for $g\to\infty$, in terms of degrees of intermediate covers, and identify each direct summand in terms of factorisation homology. For $2\le d\le 5$, we further interpret the stable rational cohomology in terms of configuration spaces with labels in partial abelian monoids; we show in particular the existence of stable odd-dimensional cohomology classes for $d=4,5$.

We construct a Hausdorff saturated directed space whose directed path space between two distinct points has nontrivial fundamental group although the corresponding trace space is a square. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from paths to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces whose compact subspaces are metrizable, these quotients are trivial q-fibrations. We construct a globular clock containing the directed circle and admitting regular maps from all q-cofibrant globular realizations, whereas a finite cellular example admits no regular map to the circle. Normalization also gives a proof of the quotient theorem for execution paths and establishes the homotopy lifting property for nondecreasing reparametrizations.

We prove that the sheaf $\infty$-topos associated to a geometric site in the sense of Lurie can be written as a partially lax limit of smaller sheaf $\infty$-topoi. This is thus a formalization of Lurie's vision for fractured $\infty$-topoi to axiomatize the relation between gros and petit topoi. As a consequence, we realize the $\infty$-topos of $\kappa$-small condensed anima as a partially lax limit of sheaf $\infty$-topoi over extremally disconnected spaces, marked at the open embeddings. Moreover, we deduce a gros version of the étale exodromy theorem.

We extend Sullivan's rational homotopy theory by constructing geometrically and algebraically minimal models for diagrams of simply connected spaces over a class of finite indexing categories. Both classes classify rational diagram homotopy types, but their minimality conditions can differ. We also show that, in contrast to classical rational homotopy theory, H-diagrams need not split into products of Eilenberg–MacLane diagrams, and characterize the indexing categories for which every H-diagram admits such a splitting. Also we apply this to computation of automorphism groups of rational homotopy types of certain diagrams of spaces.

For a given $n$-valued map, there exists a suitable minimal covering of the domain such that the composition of this covering with the map yields a split map, composed of $n$ single-valued maps, called the lift factors. Using the group action of the deck-transformation group we construct intermediate covering spaces as well as intermediate configuration spaces, allowing the original map to lift to these spaces, building a tower to illustrate at which level the lift factors split off. We then display two applications of these new tools. The first is to derive results concerning the Borsuk-Ulam property for certain coverings of the tower. The second is to obtain a sharp formula for the Nielsen number of an $n$-valued non-split map, using the coincidence number of specific single-valued maps that arise in the lifts of the tower.

We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series $R_t$ associated with the Morse-Bott inequality. These Moves preserve $R_t$, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of $R_t$ by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of $R_t$ and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of $R_t$ with the number of elementary blocks in the corresponding dimension.

We relate critical diameters of finite configurations to the topology of Vietoris-Rips complexes. For a compact metric space and $0<r<s$, we prove that the canonical inclusion from scale $r$ to scale $s$ is a homotopy equivalence whenever $[r,s)$ contains no diameter of a finite labelled configuration at which the diameter function has zero weak slope. On Riemannian manifolds, weak-slope stationarity implies Clarke criticality. At positive diameter, when the distances realizing the diameter are smooth, both are equivalent to first-order stationarity: the absence of a direction decreasing all these distances to first order. For closed connected smooth manifolds, the Clarke critical diameter spectrum has Hausdorff dimension zero, even at the cut locus. In the real-analytic case, it is finite at each fixed number of labels and countable over all label numbers. On the unit round sphere $S^m$, $m\geq1$, positive nonantipodal weak-slope stationary configurations are characterized by nonzero nonnegative equilibrium stresses. The least positive critical diameter is $\arccos(-1/(m+1))$, and the first accumulation point is $\arccos(-1/m)$. We lift every such stress through a spherical stack construction adapted from Lovász, producing stationary diameters that approach the original value from below in the next dimension as the number of layers increases. We also give example in general metric space showing that weak-slope stationary diameters need not correspond to changes in the homotopy type of Vietoris-Rips complexes.

We deformalize an LLM-generated proof within the formalism of Homotopy Type Theory that an iterated pushout construction for $\operatorname{SP}^3(X)$ is a set whenever $X$ is a set. The proof expands on a similar proof by Buchholtz for $\operatorname{SP}^2(X)$, employing a similar encode-decode strategy and the same formal machinery.

If $M$ is a closed oriented manifold of even dimension, then its $n$th symmetric product $M(n)=M^n/S_n$ is a rational homology manifold for any $n \geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thesis. We obtain a simpler result by studying the dual classes in homology, showing that their generating function (after embedding each $H_*(M(n); \mathbb{Q})$ into the Hopf algebra $H_*(M(\infty); \mathbb{Q})$) is the product of an elementary factor depending only on the Euler characteristic of $M$ and the exponential of a series consisting of odd Adams twists of the Poincaré dual of the $L$-class of $M$. We also use this to obtain both a simpler proof and a simpler expression for the cohomological $L$-classes than the previous one.

Let $\mathcal{F}^{m}=\{A\subset[m]\}$, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type $\operatorname{\v{C}ech}\left(\mathcal{F}_{\preceq a}^m; \frac{r}{2}=1\right)$ by relating it to consecutively adding vertices of $\mathbb{I}^m$, the unit hypercube of $m$th dimension, to point $0$. Further, for higher $r$, we propose a conjecture regarding contractibility of a subcomplex implying a formula for the exact number of wedges of $\mathbb{S}^ 3$'s and $\mathbb{S}^ 4$'s in the $\operatorname{\v{C}ech}$ complex.

Vietoris-Rips filtrations, which are standard in topological data analysis, are flag complexes, and a simplex tree stores these without any attaching data. In this paper we ask what survives of this economy when a flag complex $K$ is divided by a subcomplex $A$, each connected component of $A$ being crushed to a point. Such a quotient is a CW complex whose cells are the simplices of $K\setminus A$, but their attaching maps are no longer implicit. We show that for flag $K$ the face order of the quotient is strictly graded exactly when $A$ is flag, and that the surviving labelled cells are determined by those of dimension at most 3. For $m$-flag pairs the threshold is $2m+1$, and it drops to $m+2$ when $K$ is flag. The prescribed cells form a regular CW decomposition only when $A$ is full in $K$. These results justify the QF-tree: a cell table that stores, for each surviving simplex, its ordered list of $d+1$ facets with collapsed facets flagged, indexed by a trie of quotient-vertex words. For bounded dimension its size is linear in the number of surviving simplices plus the retained provenance, and we derive and verify a simple formula for the collapsed fraction above which it is smaller than the homotopy-equivalent cone model. Because a collapse changes the attaching data only on the closed star of $A$, the QF-tree can also be applied locally inside a simplex tree. For a ball-shaped $A$ in the sampled Vietoris–Rips regime the closed star is a thin shell, and the median compact budget is below the cone model at every sampled radius. An accompanying library, modelled on Gudhi, implements the QF-tree, its local variant, an editable layer with local quotient updates, gluing, disc attachment, induced maps, cup products, fundamental-group presentations and zigzag persistence, and provided experiments separate the cost of maintaining a quotient from the cost of the algebra computed on it.

This paper compares two candidate notions of morphism in the setting of fully local topological field theory. A first description can be formulated in terms of defects of codimension $k$ separating a pair of defects of codimension $(k-1)$. Lurie's cobordism hypothesis with singularities identifies such defects with suitably dualizable $k$-morphisms in the target category. A second description can be formulated in terms of symmetric monoidal (op)lax natural transformations between the truncations of the source and target theories. We show that these two descriptions agree. The comparison rests on a criterion for when a $k$-morphism determines a codimension-$k$ defect, expressed as a one-sided iterated adjunctibility condition. Assuming the ordinary cobordism hypothesis, this comparison provides a reformulation of the cobordism hypothesis for defects in terms of systems of oplax natural transformations. We also apply the criterion to the higher Morita category, where it yields a reducibility theorem for modules over $E_n$-algebras: a module over an $(n+1)$-dualizable $E_n$-algebra $A$ is $n$-dualizable over $A$ if and only if its underlying $E_{n-1}$-algebra is $n$-dualizable.

Let $P_q$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_q]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit Peterson problem of finding a minimal set of generators for $P_q$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this note, we present the minimal sets of $\mathcal{A}$-generators for $P_5$ in the cases where the degree are sufficiently general.

We study transfer systems on partially ordered sets whose Hasse diagrams are rooted trees. We prove there is a bijection between transfer systems on a tree and order-preserving functions to the natural numbers that are bounded by the rank function. We then develop a recurrence for the set of functions and use this to enumerate transfer systems on some families of trees. In addition to our study of trees, we prove for any finite poset $\mathcal{P}$, the lattice of transfer systems on the opposite poset $\mathcal{P}^{op}$ is isomorphic to the opposite lattice of transfer systems on $\mathcal{P}$. We show this by establishing a bijection between weak factorization systems and transfer systems, which builds on the results for lattices shown in previous work of Franchere, Ormsby, Osorno, Qin, and Waugh.

In this paper, we provide a construction of an acyclic discrete vector field that is compatible with a total order associated with a lower star filtration on a simplicial complex, called the colex vector field. We show that the colex vector field induces a filtered acyclic vector field, whose resulting Morse complex computes the persistent homology of the lower star filtration on the underlying simplicial complex. We show that the colex vector field can be recomputed quickly when the vertex function that induces the lower star filtration is modified via order-adjacent vertex swaps. We provide a framework for storing and computing the number of paths between cells in the simplicial complex, as well as a method to update these values quickly when the colex vector field changes. Finally, we provide publicly available proof-of-concept code for the ideas shown. When applying it to the Persistent Homology Transform, we show that its runtime is comparable to a more standard matrix reduction approach.

We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model $Y_{q+1}(n, p)$. We first give a definition of a Gaussian–semicircle law: $\mathcal{N}(0, \sigma^2)\boxplus \operatorname{SC}(s\sigma^2)$, and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian–semicircle law, in the sense of almost surely weak convergence.

Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning. They provide an axiomatically justified method to fairly distribute a model's prediction among the data features. Shapley values have not yet been applied to explain machine learning models trained on features from topological data analysis. We develop what we believe is the first such approach, focusing on the persistence landscape featurization of persistence diagrams. Because each landscape coordinate is a rank statistic, crediting a model's prediction back to individual persistent homology classes (persistence diagram points) is nontrivial. We introduce LandscapeSHAP, a method for fair credit allocation to persistence diagram points based on a model's prediction. For linear models on persistence landscapes, LandscapeSHAP has a closed form expression that gives the exact Shapley value of every persistence diagram point. In particular, there is no coalition sampling required. We further prove that the four Shapley "fairness" axioms uniquely characterize this credit allocation for any model, not only linear ones. For a general nonlinear model, this unique value can only be calculated exactly from its defining coalition averaging formula, which requires considering all $2^N$ many coalitions, where $N$ is the number of points in the persistence diagram. This is computationally intractable for persistence diagrams of realistic size. We complement the exact linear model result with an efficient Monte Carlo sampling of persistence diagram coalitions. We give convergence rates in terms of number of samples needed to approximate to a desired degree of accuracy. We also prove stability results for the LandscapeSHAP credit allocation, for any model.

We develop foundations for metastable homotopy theory using Heuts' 2-truncated Tate coalgebras.

We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence $\Omega P_{p^k}I(S^1) \simeq \mathbb{Z} \times \Omega^{\infty+2k}\tau_{>0}\mathbb{Sp}^{p^k}$ obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the Goodwillie approximations of the circle as infinite loop spaces, a phenomenon which does not hold in general. The equivalence also reduces the computation of their homotopy groups to a stable problem, making them accessible to tools such as the Adams spectral sequence and to computer calculations. We place these computations in a more general framework by developing a Goodwillie-calculus analogue of the Gray sequence, which relates the homotopy groups of successive stages of the tower.

We establish a Chouinard-type theorem for modules over algebras of cochains on the classifying spaces of a broad class of infinite groups, allowing commutative ring spectra as coefficients. We then use this result to study finiteness properties of the integral cohomology rings of certain infinite groups, as well as stratification of the corresponding module categories. In particular, we complete the (co)stratification of module categories associated to homotopical groups over fields of positive characteristic.