K-Theory and Homology
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
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Binary Leavitt algebra L_F2(1,2) has integrally acyclic general linear groups of every rank, and the Steinberg map is an isomorphism from…
· “General linear and Steinberg groups over the Leavitt algebra L_(mathbb F₂)(1,2)”
Adams operations and Stiefel connectivity yield a smooth Hermitian bundle whose frame bundle gives a global quotient.
· “Regular Bundles on Orbifolds: A Short Proof of Presentability”
Both K-groups are now explicit whenever a free abelian subgroup generates the coarse structure, purely from group data.
This representability links motives directly to Chabauty-Kim theory and gives a K-theoretic finiteness criterion for elliptic schemes
Countable K-stable algebras: product and bivariant K-theory commute, but infinite products escape the bootstrap class.
A canonical homomorphism to fppf cohomology and a characteristic-free relative Kummer sequence are constructed for faithful affine maps.
· “Relative Brauer groups and Kummer Sequence without characteristic constraint in fppf-topology”
The paper discovers that for any bicategory B equipped with G-twisting data (a strong monoidal functor Θ_A: G → Pic_B(A) for each object…
Surjectivity of the bounded comparison map settles the two conjectures named in the title.
Characterization reduces homological conjectures including finitistic dimension and Gorenstein symmetry
· “Singular equivalences of n-adjoint type and standard eventually homological isomorphisms”
Theorem A gives a concrete counterexample for all fraction fields of discrete valuation rings, settling a question left open since 1982.
· “An explicit non-elementary matrix over a bivariate Laurent polynomial ring”
A categorical construction yields topological K-theory of super Banach algebras and proves eightfold Bott periodicity.
Proof that homogeneity, cyclic symmetry, and shuffle relations reduce from the decomposition relation alone.
· “The decomposition relations alone present the Goncharov Lie coalgebra”
For even ring spectra, quotient towers by height-n ideals recover K(R) at telescope heights n+1 and above.
New exact sequences induce K-theory fibre sequences, proving resolution and Gillet-Waldhausen theorems.
Projectives are exactly the induced T(P); perfectness passes from R, heredity needs three extra conditions.
With a spectral gap, the projection's K-class depends only on generator count and the relator's cyclic factor.
A regularity condition on representation rings decides it, yielding a full corank-one classification for simple groups.
Rees algebras convert analytic-to-topological equality into a single algebraic step, packaging Getzler rescaling as a family over the line.
New construction separates bounded presentation data from a regulator-controlled singular term, realizing local K3 groups via principal…
· “Bloch-regulator Principal Parts of Cyclotomic Iwasawa Pseudomeasures”
For smooth proper varieties over finite fields, these groups are finite and field-independent.
Singular cones always leave an obstruction in NK1 or NK2; the level-2 threshold for smoothness is uniform in dimension.
· “K₂-regularity is equivalent to smoothness for toric varieties”
The characterization links dualizable additive categories to idempotent ideals, nuclear modules, and algebraic K-theory.
For Hilbert manifolds with bounded geometry, the paper constructs a homomorphism from geometric K-homology to the K-theory of the…
· “Geometric K-homology and operator K-theory for Hilbert manifolds”
Reflection functors make KK(X)_loc and the bootstrap category depend only on the underlying tree of X.
The empty summand is the generic restriction map, yielding Gersten injectivity for henselian regular local rings.
· “Finite-coefficient K-theory of henselian valued fields and Gersten injectivity”
Algebraic cycles enter only through a subgroup generated by curves, making Witt groups computable from real points.
The model's deformation solutions are exactly homotopy Rota-Baxter Lie structures.
· “Deformations and homotopy theory of Rota-Baxter Lie algebras”
Trivial extension of the Xu–Snashall algebra has Hochschild cohomology modulo nilpotence that is not finitely generated.
· “A Symmetric Counterexample to the Snashall--Solberg Conjecture”
Subgroups of p-divisible groups yield Galois extensions on geometric fixed points, recovering KU and TMF actions.
The C_p-fixed homotopy ring contains divided powers and an exterior class, reshaping equivariant periodicity and future trace-method…
· “Computations in Equivariant Topological Hochschild Homology”
Even over the sphere spectrum, the infinity-category of localizing motives has no compact generators.
Motivic stable homotopy theory gains a six-functor formalism for the Hecke stack.
The result fills a gap in the literature and settles the exact multiplicative strength of the filtration.
Nisnevich descent plus affine-line invariance make continuous K-theory a motivic object, with condensed-spectra formulas.
· “Representability of continuous K-theory in rigid analytic motivic mathbb{A}¹-homotopy theory”
An explicit nonzero class in K_2(A;Z/3) vanishes over the fraction field, so finite-coefficient Gersten injectivity fails.
· “Finite-coefficient Gersten injectivity fails in ramified mixed characteristic”
Truncated polynomial rings over number fields now have a closed-form integral motivic cohomology.
Homological Real trace methods are developed, yielding spectral sequences that compute the continuous mod two Bredon homology of Real…
The same integer counts Weyl cosets, packet members, and compact-dual Euler characteristic — determining which primes block integral splitti
· “L-packet multiplicity and integral structure in the K-theory of real inner forms”
Proof that spectra, Berry curvature and quantum metric vary while the topological class stays fixed.
· “Beyond K-Theory: Geometry and Holomorphy in Hyperbolic Band Theory”
Where Stickelberger values vanish, first derivatives take over and annihilate the even K-groups.
· “A Coates-Sinnott-type Theorem for First Derivatives of Artin L-Functions”
A 3D textile framework reduces the hardest axiom of 3-graph theory to two commutative diagrams, and its homology matches.
· “Higher-Dimensional Symbolic Dynamics: A Textile Framework For 3-graphs”
Classical count of 10 lines upgraded to a quadratic-form-valued invariant that works over any perfect field.
· “A quadratically enriched count of lines on a smooth del Pezzo surface of degree 5”
This objectwise test yields flat model structures on Q-shaped derived categories and describes their cofibrant objects.
· “Flat models for Q-shaped derived categories via PGF objects”
Only the strong quotients can control quantized macroscopic observables
On complex quadrics, every sphere homotopy class is algebraic; on even-cellular varieties, bundles match exactly.
The condition equips K_f with Chern classes, a λ-ring structure, and Serre duality — with applications to matroids and toric bundles.
A single weighted intersection count ψ governs how many regions curves make, the shape of their network, and the Hodge structure of the comp
· “From Regions to Hodge Structures: The Topological Study of semialgebraic Curves Configurations”
Odd groups follow Witt-vector formulas; even groups vanish over algebraically closed fields.
A protected corner theorem rules out the all-infinite case, leaving a single-infinite shape or the mutual dumbbell and pinning the Euler cha
Arbitrary finite-dimensional coefficients let representation homology expose singularities that matrix coefficients miss.
· “Derived representation schemes with arbitrary coefficients and associative smoothness”
A covering's nontrivial first homology proves the embedded line cannot be straightened by a coordinate change.
A short exact sequence reduces TK1 of tame division algebras to the graded case plus p-primary central roots of unity.
A single homology theory detects invertibility, weights, and nilpotence in the finite cellular category.
· “Picard groups and composition nilpotence for finite cellular isotropic spectra”
For isolated klt-type varieties, the next-to-bottom rational K-group is the weight-two Hodge part of middle cohomology.
A classifying diagram for double categories with shared isomorphisms identifies them
· “Recognizing CGW categories among pointed stable double Segal spaces”
A functorial replacement lands in projective spectra and preserves cofibrations
The long exact sequence becomes a sequence of lambda-homomorphisms, preparing refined Riemann–Roch theorems
The last case of the Morel structure conjecture now holds for every perfect field
In dimensions three and below that is the whole story; dimension four needs one extra secondary obstruction.
· “Algebraizability of Vector Bundles over Real Algebraic Varieties”
The central discovery is that the full Hopf algebra structure on F(Z) is not needed to recover the derived functor LF on coconnected…
Langlands-dual sectors are homotopy equivalent except when spheres meet projective spaces, settling a conjecture with a dyadic refinement.
A single conjugation–duality doubling construction yields 8-periodic sequences in quadratic and hermitian form theory.
Formula linking cap products and cyclotomic maps on THH recovers classical Cartier isomorphism and computes quantum Steenrod operations viap
They compute relative homology of partial Borel–Serre compactifications for GL_n
· “Presenting restricted Steinberg modules of general linear groups”
Odd-codimension definition plus explicit values for Kronecker, fibration and orbifold cases
The computation over complex numbers yields a motivic analog of Bökstedt periodicity via tau comparisons and power operation relations.
· “Motivic Hochschild homology of mod 2 motivic cohomology over algebraically closed fields”
The equivariant isomorphism connects sphere homology from simplicial cones to trace-fixed Milnor K-theory structures over the rationals.
Exposition of the Drinfeld-Lau construction applies on rings whose reductions are perfect F_p-algebras.
Matrices for dimensions up to 3 and a recipe for all higher dimensions turn abstract classes into concrete operators.
· “The real K-theory of the sphere with an arbitrary involution”
Spectral flow of real Fredholm operators on Hilbert C*-modules equals a Fredholm index via Van Daele K-theory.
· “Analytic index theory and spectral flow in real Hilbert C^*-modules”
The construction supplies the E2-term of a multiplicative Adams-type spectral sequence to p-completed homotopy groups.
· “Continuous homology of topological periodic homology of complex cobordism”
Self-contained proof shows curvature and angle conditions force three-dimensional manifolds to be rigid.
· “Gromov's dihedral rigidity conjecture in dimension three”
Bivariant algebraic K-theory respects Morita equivalence for G-algebras and G-graded algebras, including crossed-product equivalences under
Uniform construction for every N yields regulators asymptotic to 3/2 zeta(3) N^2 and checks L-value predictions at s=3 for N=3,4,6.
Functoriality of cross-sectional algebras allows the map for Cartan pairs and non-Hausdorff groupoids.
· “A Baum-Connes assembly map for essential semigroup crossed products”
Symbolic descriptions for rational K4(3) and indecomposable K5(3) of fields extend known low-weight cases.
Wilson spaces control orientations; the results compute Real THH of KUR and MUPR
Independent resolutions then reduce homology and K-theory computations for ample groupoids to the discrete case.
· “Discretisation and independent resolutions of ample groupoids”
It also gives a natural connection on the periodic cyclic homology of OCHA families, flat up to chain homotopy.
· “Non-commutative calculus and Getzler-Gauss-Manin connections for Open-closed Homotopy Algebras”
The higher groups reduce to the Sylow p-subgroup ring k[C_p^r] and are obtained from topological cyclic homology for any perfect field of ch
· “The algebraic K-theory of k[operatorname{SL}₂(mathbb{F}_q)]”