67 papers
This paper clarifies that the invariance of -theory under duality is a purely formal property, contrasting it with the non-formal nature of the general statement for arbitrary localizing invariants, and provides a counterexample to the claim that the universal localizing invariant is invariant under taking opposite categories.
This paper introduces the framework of "fat Lie theory" to establish a one-to-one correspondence between fat extensions and abstract 2-term representations up to homotopy, while demonstrating that fat extensions of groupoids correspond to general linear PB-groupoids and core extensions to vertically/horizontally core-transitive double groupoids, thereby upgrading existing correspondences to equivalences of categories.
This paper establishes a natural -multicategory structure on moduli spaces of pseudo-holomorphic polygons to provide a uniform operadic framework for various -type structures, including Fukaya categories, as algebras over differential graded -multicategories.
This paper presents a simplified approach to the stable Hopf invariant by offering elementary proofs of the Cartan, Composition, and Transfer formulas, and extends these results to the stable category of -spaces for discrete groups.
This paper introduces the persistent local Laplacian formalism, a theoretically grounded and highly parallelizable framework that overcomes the sensitivity and scalability limitations of traditional topological data analysis by establishing a generalized persistent Hodge isomorphism and unitary equivalence to efficiently extract fine-grained local structural signatures from large-scale datasets.
This paper establishes a combinatorial sufficient condition based on multinet structures for the Milnor fiber of a complex hyperplane arrangement to be non-1-formal and uses this criterion to construct an infinite family of monomial arrangements with non-formal Milnor fibers.
This paper characterizes the fundamental group of disjointly tree-graded spaces in terms of the fundamental groups of their constituent pieces, establishing an embedding into an inverse limit of free products even when local simple connectivity is absent.
This paper extends the theory of ambidexterity by proving that the norm square induced from a weakly ambidextrous morphism commutes under Beck-Chevalley fibrations, thereby generalizing the naturality of the norm established by Hopkins and Lurie and recovering specific results on local systems and equivariant powers by Carmeli, Schlank, and Yanovski.
This paper establishes a folklore theorem characterizing the embeddability of partial groups in groups via unique parenthesization of words, investigates examples of non-embeddable cases, and proves that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.
This paper establishes a comprehensive -operadic framework for embedding calculus by analyzing towers of truncated right-modules over unital -operads to generalize Goodwillie-Weiss theory to bordism categories, derive new variants for topological and configuration embeddings, and prove key results on convergence, delooping, and homology 4-spheres.
This paper constructs a spectral sequence converging to operadic tangent cohomology by utilizing filtrations from towers of cofibrations, thereby providing new algebraic descriptions of the Serre spectral sequence and the rational homotopy groups of self-fiber-homotopy equivalences.
This paper investigates -categories by comparing the -categories obtained via core and localization functors in the limit , demonstrating that the latter is a reflective localization of the former while also exploring intermediate localizations arising from coinductive notions of invertibility.
This paper computes the -graded Bredon cohomology of equivariant universal and classifying spaces with constant coefficients, providing an explicit description of the resulting coefficient ring and applying these results to study lifts of cohomology operations via equivariant complex projective spaces.
This paper demonstrates that the homotopy fixed points spectrum of higher topological Hochschild homology applied to a commutative ring spectrum detecting -elements yields a spectrum detecting -elements with , thereby establishing a mechanism for higher chromatic redshift beyond the single-step increase typically observed in algebraic K-theory.
This paper reviews the concept of Kalinin effectivity, establishes that wonderful compactifications of hyperplane arrangements and configuration spaces derived from Kalinin effective manifolds retain this property, and applies these results to demonstrate the effectivity of the Deligne-Mumford space of real rational curves and to investigate Smith-Thom maximality for Hilbert squares.
This paper investigates the homotopy types and connectivity of total cut complexes and their Alexander duals for various graph families, including powers of cycles, complete multipartite graphs, and Cartesian products, thereby resolving specific conjectures regarding these structures.
This paper establishes that the homotopy type of the independence complex of a generalized Mycielskian graph is determined by the homotopy types of the independence complexes of the original graph and its Kronecker double cover, and applies this result to compute these types for paths, cycles, and categorical products of complete graphs.
This paper investigates the local-triviality dimensions of actions on -algebras within noncommutative Borsuk-Ulam theory, demonstrating that free actions do not necessarily possess finite weak local-triviality dimensions and that these invariants can exhibit discontinuity or exceed fiber values in continuous fields, while establishing conditions for upper semicontinuity through examples involving noncommutative tori and spheres.
This paper proves that the Quillen posets of -extensions of simple unitary groups have non-zero homology in the largest possible dimension, thereby establishing the 1992 Aschbacher-Smith conjecture and confirming Quillen's conjecture for odd primes.
This paper establishes a sharp topological dichotomy for slices of the Sierpinski tetrahedron, demonstrating that slices at dyadic rational heights possess finitely many connected components with infinite first Čech homology, whereas slices at non-dyadic rational heights are totally disconnected with vanishing positive-degree homology.