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We prove an $L^p$-version of the coarse Baum--Connes conjecture for spaces that coarsely embedds into $\ell^q$-spaces for any $p$ and $q$ in $[1,\infty)$., Comment: 27 pages

Several candidates for a motivic spectrum representing hermitian K-theory in the Morel-Voevodsky motivic stable homotopy category over schemes in which 2 is not necessarily invertible exist. This note shows that the cellular absolute motivic spectrum constructed in the thesis of the first author via the geometry of orthogonal and hyperbolic Grassmannians over any scheme coincides with the motivic ring spectrum constructed recently by Calm\`es, Harpaz, and Nardin., Comment: 8 pages

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the subcategory of dualizable objects in the derived category of $I$-complete complexes of $R$-modules. Our criterion is always satisfied when $R$ is noetherian. When specialized to $R$ local and noetherian and to $I$ the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause and Pevtsova., Comment: 20 pages

We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series., Comment: 25 pages, comments are welcome

We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$. We relate these groups to the values of the non-abelian derived functors of the functor $M \mapsto (M \otimes_{\mathbb{Z}/4} M)^{C_2}$ at the $\mathbb{Z}/4$-module $\mathbb{Z}/2$, which we precisely calculate with computer assistance up to degree $6$, and calculate in general up to slight remaining ambiguity. Using these results we compute $\pi_i(\mathrm{TCR}(\mathbb{Z}/4)^{\phi \mathbb{Z}/2})$ exactly for $i \le 1$, up to an extension problem for $2 \le i \le 5$, and describe the asymptotic growth of this group for large $i$. A consequence of these computations is that there exists some $0 \le i \le 5$ such that the canonical map comparing the genuine symmetric and symmetric $L$-theory spectra of $\mathbb{Z}/4$ is not an isomorphism on degree $i$ homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees., Comment: 44 pages

We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy of possible commutativity conditions in terms of progressive stabilization of these data. We also show that monoidal structures on bicategories give rise to biextensions of a pair of (abelian) groups by a Picard groupoid, and that the progressive vanishing of obstructions determined by the tower of commutative structures corresponds to appropriate symmetry conditions on these biextensions. In the fully symmetric case, which leads us fully into the stable range, we show how our computations can be expressed in terms of the cubical Q-construction underlying MacLane (co)homology., Comment: 27 pages. Preliminary version, comments welcome!

We exhibit a canonical equivalence between the hermitian $K$-theory (alias Grothendieck-Witt) spectrum of an exact form category and that of its derived Poincar\'e $\infty$-category, with no assumptions on the invertibility of $2$. Along the way, we obtain a model for the nonabelian derived functor of a nondegenerate quadratic functor on an exact category., Comment: Comments welcome!

We establish conditions under which an inclusion of finitely aligned left-cancellative small categories induces inclusions of twisted C*-algebras. We also present an example of an inclusion of finitely aligned left-cancellative monoids that does not induce a homomorphism even between (untwisted) Toeplitz algebras. We prove that the twisted C*-algebras of a jointly faithful self-similar action of a countable discrete amenable groupoid on a row-finite k-graph with no sources, with respect to homotopic cocycles, have isomorphic K-theory., Comment: 18 pages

Consider a proper action of $\mathbb{Z}^d$ on a smooth (perhaps non-paracompact) manifold $M$. The $p^{th}$ cohomology $H^p(\mathbb{Z}^d,\ \Gamma_{\mathrm{c}}(\mathcal{F}))$ valued in the space of compactly-supported sections of a natural sheaf $\mathcal{F}$ on $M$ (such as those of smooth function germs, smooth $k$-form germs, etc.) vanishes for $p\ne d$ (the cohomological dimension of $\mathbb{Z}^d$) and, at $d$, equals the space of compactly-supported sections of the descent ($\mathbb{G}$-invariant push-forward) $\mathcal{F}/\mathbb{Z}^d$ to the orbifold quotient $M/\mathbb{Z}^d$. We prove this and analogous results on $\mathbb{Z}^d$ cohomology valued in $\Phi$-supported sections of an equivariant appropriately soft sheaf $\mathcal{F}$ in a broader context of $\mathbb{Z}^d$-actions proper with respect to a paracompactifying family of supports $\Phi$, in the sense that every member of $\Phi$ has a neighborhood small with respect to the action in Palais' sense., Comment: 20 pages + references

We prove basic statements about the Hermitian K-theory of exact form categories with weak equivalences. Notably, we extend a quadratic functor with values in abelian groups from an exact category to its category of bounded chain complexes in a way that does not change Grothendieck-Witt spaces. This is used in joint work with Marlowe for the comparison of the classical 1-categorical version of the Hermitian K-theory of exact categories with the infinity-categorical version of Calmes-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle.

We develop a correspondence between presentations of compactly generated triangulated categories as localizations of derived categories of ring spectra and proxy-small objects, and explore some consequences. In addition, we give a characterization of proxy-smallness in terms of coproduct preservation of the associated corepresentable functor `up to base change'., Comment: 15 pages

In this paper we are going to compute the $ \mathrm{KW} $-Euler classes for rank 2 vector bundles on the classifying stack $ \mathcal{B}N $, where $N$ is the normaliser of the standard torus in $SL_2$ and $\mathrm{KW}$ represents Balmer's derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove K\"unneth formulas for products of $GL_n$'s and $SL_n$'s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in $SL$-oriented theories.

We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite colimits. In particular, when $\mathscr{C}$ is a Mal'tsev variety of universal algebras, the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is also a Mal'tsev variety, of which we describe the corresponding algebraic theory. When $\mathscr{C}$ is a naturally Mal'tsev category, the reflector from $ \mathrm{Grpd}^2(\mathscr{C}) $ to 2-$ \mathrm{Grpd}(\mathscr{C}) $ has an additional property related to the commutator of equivalence relations. We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is semi-abelian when $\mathscr{C}$ is semi-abelian, and then provide sufficient conditions for 2-$ \mathrm{Grpd}(\mathscr{C}) $ to be action representable., Comment: 12 pages

We introduce a general definition of a $n$-crossed module of $P$-algebras over an algebraic operad $P$, which coincides with historical definitions in the cases of the operads As and Lie and $n = 1$. We establish a natural isomorphism between the abelian group of equivalence classes of $n$-crossed modules over a pair $(A,M)$ for an operad $P$ and the $(n+1)^\text{th}$ operadic cohomology group of $A$ with coefficients in $M$., Comment: 21 pages

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace invariance. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods.

We develop a cohomological approach to M\"obius inversion using derived functors in the enriched categorical setting. For a poset $P$ and a closed symmetric monoidal abelian category $\mathcal{C}$, we define M\"obius cohomology as the derived functors of an enriched hom functor on the category of $P$-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical M\"obius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota's Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra., Comment: 23 pages

Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity. To prove this, we study some analogue of Bloch's higher Chow groups and establish for this complex an analogue Beilinson-Soule vanishing conjecture. A. Goncharov defined a group of functional equations for classical polylogarithms. We show that any such functional equation formally follows from functional equations for Chow polylogarithms stated above.

Any closed connected Riemannian spin manifold of non-zero $\hat{A}$-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that any closed connected Ricci-flat spin manifold of dimension $\geq 2$ with non-vanishing Rosenberg index has special holonomy., Comment: 8 pages, comments welcome

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\Theta \to \Phi$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\Theta \to \Phi$. The previous case corresponded to $\Theta = \Phi = \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\Theta$ and $\Phi$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $\Omega^{(\Theta,\Phi),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest., Comment: Comments welcome! v2: updated references and other minor changes

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology and its relation with the relative notion of fundamental group. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically closed field of characteristic zero, such that $Q_B$ is a subquiver of $Q_A$. We show that if the complement of $Q_A$ by the arrows of $Q_B$ is a simple directed graph, then the first relative Hochschild cohomology $\text{HH}^1(A|B)$ is a solvable Lie algebra. We also compute the Lie algebra structure of the first relative Hochschild cohomology for radical square zero algebras and for dual extension algebras of directed monomial algebras. Finally, we introduce the notion of fundamental group for a pair of an algebra $A$ and a subalgebra $B$ and we construct the relative version of the map from the dual fundamental group into the first Hochschild cohomology., Comment: 28 pages

We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter $\delta$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $\delta$ and integer $p \geq 0$ this gives operator system invariants $\mathcal V_p^\delta(-,n)$, indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the $K_p^\delta$-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either $K_0^\delta$ or $K_1^\delta$. We illustrate our invariants by means of the spectral localizer., Comment: 14 pages, 2 figures

A Koszul duality-type correspondence between coderived categories of conilpotent differential graded Lie coalgebras and their Chevalley-Eilenberg differential graded algebras is established. This gives an interpretation of Lie coalgebra cohomology as a certain kind of derived functor. A similar correspondence is proved for coderived categories of commutative cofibrant differential graded algebras and their Harrison differential graded Lie coalgebras., Comment: 23 pages

We interpret the coarse symbol and index class of a Callias type Dirac operator $D+\Psi$ on a manifold $M$ as a pairing between the coarse symbol and index classes associated to $D$ and K-theory classes of the coarse corona of $M$ or $M$ itself determined by $\Psi$. Local positivity of $D$ and local invertibility of $\Psi$ are incorporated in terms of support conditions on the $K$-theoretic level., Comment: 106 pages

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this description of $\mathrm{KK}^\mathcal{C}$ to prove the stable uniqueness theorem in this setting.

We generalise the classical Bass-Heller-Swan decomposition for the K-theory of (twisted) Laurent algebras to a splitting for general localising invariants of certain categories of twisted automorphisms. As an application, we obtain splitting formulas for Waldhausen's A-theory of mapping tori and for the K-theory of certain tensor algebras. We identify the Nil-terms appearing in this splitting in two ways. Firstly, as the reduced K-theory of twisted endomorphisms. Secondly, as the reduced K-theory of twisted nilpotent endomorphisms. Finally, we generalise classical vanishing results for Nil-terms of regular rings to our setting., Comment: 30 pages. Comments welcome!

In this article we study a relative monoidal version of the Bondal-Orlov reconstruction theorem. We establish an uniqueness result for tensor triangulated category structures $(\boxtimes,\mathbb{1})$ on the derived category $D^{b}(X)$ of a variety $X$ which is smooth projective and faithfully flat over a quasi-compact quasi-separated base scheme $S$ in the case where the fibers $X_{s}$ over any point $s\in S$ all have ample (anti-)canonical bundles. To do so we construct a stack $\Gamma$ of dg-bifunctors which parametrize the local homotopical behaviour of $\boxtimes$, and we study some of its properties around the derived categories of the fibers $X_{s}$.

We use homotopy theoretic ideas to study the $K$-theory of (graded, Real) $C^*$-algebras in detail. After laying the foundations, and deriving the formal properties, the comparison of the model with the Kasparov picture of $K$-theory has been made, and Bott periodicity has been proven using a Dirac-dual Dirac method., Comment: This article is roughly the content of the author's masters thesis written under the supervision of Prof. Dr. Johannes Ebert at the University of Muenster in 2023. An updated version will be available at a later date

We prove that for any presentably symmetric monoidal $\infty$-category $\mathcal{V}$, the $\infty$-category $\mathbf{Mod}_\mathcal{V}(\mathbf{Pr}^{\mathrm{L}})^{\mathrm{dbl}}$ of dualizable presentable $\mathcal{V}$-modules and internal left adjoints between them is itself presentable. Along the way, we survey formal properties of these dualizable $\mathcal V$-modules. We pay close attention to the case of the $\infty$-category of spectra, where we survey the foundational properties of ``compact morphisms''., Comment: 80 pages; comments welcome!

We survey the theory of locally rigid and rigid symmetric monoidal $\infty$-categories over an arbitrary base $\mathcal{V}\in\mathrm{CAlg}(\mathbf{Pr}^\mathrm{L})$. Along the way, we introduce and study ``$\mathcal{V}$-atomic morphisms'', which are analogues of compact morphisms over an arbitrary base $\mathcal{V}$., Comment: 59 pages, comments welcome!

We study a weight-exact localization pi of a well generated triangulated category C along with the embedding of the hearts of adjacent t-structures coming from the functor right adjoint to pi. We prove that the functors relating the corresponding four hearts are completely determined by the heart Hw of the weight structure on C along with the set of Hw-morphisms that we invert via pi; it also suffices to know the corresponding embedding of the hearts of t-structures. Our results generalize the description of non-commutative localizations of rings in terms of weight-exact localizations given in an earlier paper of the first author. That paper was essentially devoted to weight-exact localizations by compactly generated subcategories, whereas in the current text we focus on "more complicated" localizations. We recall that two types of localizations of the sort we are interested in were studied by several authors. They took C=D(R-mod); the heart of the first t-structure was equivalent to R-mod, and the second heart was equivalent to the exact abelian category $U_{contra}\subset R-mod$ of U-contramodules (corresponding to a set of Proj R-mod-morphisms U related either to a homological ring epimorphism $u:R\to UU$ or to an ideal of I of R). The functor $R-mod\to U_{contra}$ induced by pi is a certain completion one. Consequently, the hearts of the corresponding weight structures are equivalent to Proj R-mod and to the subcategory of projective objects of U_{contra}, respectively. Moreover, the connecting functors between these categories are isomorphic to ones coming from any weight structure class-generated by a single compact object whose endomorphism ring is R^{op}; in particular, one can take R=Z and C=SH and re-prove some important statements due to Bousfield., Comment: 28 pages. Comments are welcome!

We develop important properties of the KK-functor on the basis of split exactness., Comment: 12 pages

We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the trivial left and right module all coincide. We also provide convenient criteria for smoothness and the twisted Calabi-Yau property for such braided Hopf algebras (without the cosemisimplicity assumption on $H$), in terms of properties of the trivial module.These generalize a well-known result in the case of ordinary Hopf algebras. As an illustration, we study the case of the coordinate algebra on the two-parameter braided quantum group $\textrm{SL}_{2}$.

We generalize several basic facts about the motivic sphere spectrum in $\mathbb A^1$-homotopy theory to the category $\mathrm{MS}$ of non-$\mathbb A^1$-invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace $\mathbf 1_\mathbb Q^+$ of the rational motivic sphere is the rational motivic cohomology spectrum $\mathrm H\mathbb Q$, which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of $\mathrm H\mathbb Q$-modules in $\mathrm{MS}$: a rational motivic spectrum is an $\mathrm H\mathbb Q$-module iff it is orientable, iff the involution $\langle -1\rangle$ is the identity, iff the Hopf map $\eta$ is zero, iff it satisfies \'etale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over $\mathbb Z[\zeta_n]$ for any $n\geq 3$., Comment: 13 pages. Comments welcome!

Given a smooth scheme $X$ over the field $\mathbb{R}$ of real numbers and a line bundle $\mathcal{L}$ on $X$ with associated topological line bundle $L=\mathcal{L}(\mathbb{R})$, we study the real cycle class map $\widetilde{\gamma}_{\mathbb{R}}:\widetilde{\mathrm{CH}}^c(X,\mathcal{L})\rightarrow\mathrm{H}^c(X(\mathbb{R}),\mathbb{Z}(L))$ from the $c$-th Chow-Witt group of $X$ to the $c$-th cohomology group of its real locus $X(\mathbb{R})$ with coefficients in the local system $\mathbb{Z}(L)$ associated with $L$. We focus on the cases $c\in\{0,d-2,d-1,d\}$ where $d$ is the dimension of $X$ and we formulate an analogue of the Hodge conjecture in terms of the exponents of the cokernel of $\widetilde{\gamma}_{\mathbb{R}}$ that is corroborated by the results obtained in those codimensions., Comment: 21 pages. Comments welcome!

For any given submersion $\pi:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure $\boldsymbol{\pi}$, we construct an analytic index $\textrm{ind}^a_k$ in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory on the cocycle level by associating to every cocycle $(\mathbf{E}, \mathbf{F}, \alpha)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $X$ a cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, \alpha)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $B$. The cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, \alpha)$ is defined in terms of the twisted spin$^c$ Dirac operators associated to $(\mathbf{E}, \boldsymbol{\pi})$ and $(\mathbf{F}, \boldsymbol{\pi})$, which are not assumed to satisfy the kernel bundle assumption. We prove a Riemann-Roch-Grothendieck type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory, which expresses the Cheeger-Chern-Simons form of $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, \alpha)$ in terms of that of $(\mathbf{E}, \mathbf{F}, \alpha)$. Furthermore, we show that the analytic index $\textrm{ind}^a_k$ and the Riemann-Roch-Grothendieck type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory refine the geometric bundle part of the analytic index and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory, respectively. An as intermediate result, we give a proof that the analytic indexes in differential $K$-theory defined without the kernel bundle assumption via the Atiyah-Singer-Gorokhovsky-Lott approach and the Miscenko-Fomenko-Freed-Lott approach, respectively, are equal., Comment: 60 pages. Submitted for publication. Comments are very welcome

A field $K$ is $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ completely discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of such $K$ with many coefficients, including finite coefficients of any order. Previously, such duality was only known in few cases : as a perfect pairing of finite groups for finite coefficients prime to $\mathrm{char} k_0$ in general, or for any finite coefficients when $k_1$ is $p$-adic ; or as a perfect pairing of locally compact Hausdorff groups for the $\mathrm{fppf}$ cohomology of finite group schemes when $K$ is local. With no obvious reasonable topology available, we abandon perfectness altogether and instead obtain nondegenerate pairings of abstract abelian groups. This is done with new diagram-chasing results for pairings of torsion groups, allowing an approach by d\'evissage which breaks our results down to the study of pairings $K^M_r(K)/p\times H^{d+1-r}_p(K)\to\mathbb Z/p$ using results of Kato., Comment: 48 pages, comments welcome !

Given a discrete group $G$ with a finite model for $\underline{E}G$, we study $K(n)^*(BG)$ and $E^*(BG)$, where $K(n)$ is the $n$-th Morava $K$-theory for a given prime and $E$ is the height $n$ Morava $E$-theory. In particular we generalize the character theory of Hopkins, Kuhn and Ravenel who studied these objects for finite groups. We give a formula for a localization of $E^*(BG)$ and the $K(n)$-theoretic Euler characteristic of $BG$ in terms of centralizers. In certain cases these calculations lead to a full computation of $E^*(BG)$, for example when $G$ is a right angled Coxeter group, and for $G=SL_3(\mathbb{Z})$. We apply our results to the mapping class group $\Gamma_\frac{p-1}{2}$ for an odd prime $p$ and to certain arithmetic groups, including the symplectic group $Sp_{p-1}(\mathbb{Z})$ for an odd prime $p$ and $SL_2(\mathcal{O}_K)$ for a totally real field $K$., Comment: 50 pages

We outline how the group $K_{-1}( \mathbb{Z}[G] )$ for a finite group $G$ can be computed using the computer language $GAP$ and compile a table of all groups $G$ of order less than $100$ that have torsion in $K_{-1}( \mathbb{Z}[G] )$.

For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the functor. For half-exact functors, the obtained sequences are exact. For general functors, nontrivial homology may only appear at the derived functors. Specializing to the familiar Hom and tensor product functors on finitely presented modules, we recover the classical formulas of Auslander. Unlike those formulas, our results hold for arbitrary rings and arbitrary modules, finite or infinite. The same formalism leads to universal coefficient theorems for homology and cohomology of arbitrary complexes. The new results are even more explicit for the cohomology of projective complexes and the homology of flat complexes., Comment: 29 pages

We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of presheaves. We also show that it is a crossed module over $G$ in a unique way, in particular, that the $\mathrm K_2$-functor is central. If $G$ is globally isotropic in a suitable sense, then the Steinberg group functor exists as an ordinary group-valued functor and all such abstract Steinberg groups are crossed modules over the groups of points of $G$.

Given a self-adjoint operator $H\geq 0$ and (appropriate) densely defined and closed operators $P_{1},\dots, P_{n}$ in a Hilbert space $\mathscr{H}$, we provide a systematic study of bounded operators given by iterated integrals \begin{align}\label{oh} \int_{\{ 0\leq s_1\leq \dots\leq s_n\leq t\}}\mathrm{e}^{-s_1H}P_{1}\mathrm{e}^{-(s_2-s_1)H}P_{2}\cdots \mathrm{e}^{-(s_n-s_{n-1})H}P_{n} \mathrm{e}^{-(t-s_n)H}\, \mathrm{d} s_{1} \ldots \mathrm{d} s_{n},\quad t>0. \end{align} These operators arise naturally in noncommutative geometry and the geometry of loop spaces. Using Fermionic calculus, we give a natural construction of an enlarged Hilbert space $\mathscr{H}^{(n)}$ and an analytic semigroup $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ thereon, such that $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ composed from the left with (essentially) a Fermionic integration gives precisely the above iterated operator integral. This formula allows to establish important regularity results for the latter, and to derive a stochastic representation for it, in case $H$ is a covariant Laplacian and the $P_{j}$'s are first-order differential operators. Finally, with $H$ given as the square of the Dirac operator on a spin manifold, this representation is used to derive a stochastic refinement of the Duistermaat-Heckman localization formula on the loop space of a spin manifold.

We construct a fundamental theory of the derived category of non-finite bi-filtered complexes., Comment: 30 pages

In this paper, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces which refines the coarse Baum-Connes conjecture. And we prove that QCBC is derived by the coarse Baum-Connes conjecture with coefficients which provides many examples satisfying QCBC. In the end, we show QCBC can be reduced to the uniformly quantitative coarse Baum-Connes conjecture with coefficients of a sequence of bounded metric spaces., Comment: 30 pages; all comments are welcome! arXiv admin note: text overlap with arXiv:2410.11662

Inspired by the quantitative $K$-theory, in this paper, we introduce the coarse Baum-Connes conjecture with filtered coefficients which generalizes the original conjecture. The are two advantages for the conjecture with filtered coefficients. Firstly, the routes toward the coarse Baum-Connes conjecture also work for the conjecture with filtered coefficients. Secondly, the class of metric spaces that satisfy the conjecture with filtered coefficients is closed under products and yet it is not known for the original conjecture. As an application, we show some new examples of product metric spaces for the coarse Baum-Connes conjecture., Comment: 22 pages; all comments are welcome!

We give a Borel-type presentation of the torus-equivariant (small) quantum $K$-ring of flag manifolds of type $C$., Comment: 40 pages

It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise., Comment: 32 pages, two Main Theorems (A,B) added, comments welcome!

We provide an explicit computation of the topological $K$-theory groups $K_*(C_r^*(\mathbb{Z}^n\rtimes \mathbb{Z}/m))$ of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}Z/m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\Z/m$-action on $\mathbb{Z}^n$. This generalizes previous computations of L\"uck-Davis and Langer-L\"uck., Comment: 10 pages. Comments are welcome

In this paper, we prove an upper bound on the $\widehat{A}$ genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral analysis of the Dirac operator. We also construct an example to show that the Neumann isoperimetric constant in this bound is necessary. Our result partially answers a question of Gromov on bounding characteristic numbers using scalar curvature lower bound., Comment: Comments are welcome!

We build a hermitian Q-construction that extends that of Calm\`es et al, and compare it with the real S-construction of the current authors. From this, we deduce an equivalence between the real K-theory genuine $C_2$-spaces of the aforementioned works.

The elementary action of symplectic and orthogonal groups on unimodular rows of length $2n$ is transitive for $2n \geq \max(4, d+2)$ in the symplectic case, and $2n \geq \max(6, 2d+4)$ in the orthogonal case, over monoid rings $R[M]$, where $R$ is a commutative noetherian ring of dimension $d$, and $M$ is commutative cancellative torsion free monoid. As a consequence, one gets the surjective stabilization bound for the $K_1$ for classical groups. This is an extension of J. Gubeladze's results for linear groups., Comment: Transformation Groups (2024)