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George Raptis

Publications and source records attributed to George Raptis.

At least 19 recordsLinked to original sources
arXiv2026

The span-squares adjunction

We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. Using this adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $Q$-, the $S$-, the cobordism model, and the squares construction.

math.CT↗
arXiv2025

Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $\pi_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $\pi_1$ the Kervaire semicharacteristic.

math.AT↗
arXiv2025

The Serre spectral sequence in bounded cohomology

We construct the analogue of the Serre spectral sequence for the bounded cohomology of simplicial sets with seminormed local coefficients. As applications, we obtain a (non-isometric) generalization of Gromov's mapping theorem and some partial results on the simplicial volume of manifold bundles.

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arXiv2024

Some remarks on acyclicity in bounded cohomology

We show that a surjective homomorphism $\varphi \colon \Gamma \to K$ of (discrete) groups induces an isomorphism $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \varphi^{-1} V)$ in bounded cohomology for all dual normed $K$-modules $V$ if and only if the kernel of $\varphi$ is boundedly acyclic. This complements a previous result by the authors that characterized this class of group homomorphisms as bounded cohomology equivalences with respect to $\mathbb{R}$-generated Banach $K$-modules. We deduce a characterization of the class of maps between path-connected spaces that induce isomorphisms in bounded cohomology with respect to coefficients in all dual normed modules, complementing the corresponding result shown previously in terms of $\mathbb{R}$-generated Banach modules. The main new input is the proof of the fact that every boundedly acyclic group $\Gamma$ has trivial bounded cohomology with respect to all dual normed trivial $\Gamma$-modules.

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arXiv2022

Amenability and Acyclicity in Bounded Cohomology Theory

Johnson's characterization of amenable groups states that a discrete group $Γ$ is amenable if and only if $H_b^{n \geq 1}(Γ; V) = 0$ for all dual normed $\mathbb{R}[Γ]$-modules V. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective group homomorphism $ϕ\colon Γ\to K$ has amenable kernel H if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(Γ; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[Γ]$-module V. In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms $ϕ\colon Γ\to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all Banach $Γ$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms, that is, the class of group homomorphisms $ϕ\colon Γ\to K$ for which the restriction maps in bounded cohomology $H^\bullet_b(K; V) \to H^\bullet_b(Γ; ϕ^{-1}V)$ are isomorphisms for a suitable family of dual normed $\mathbb{R}[K]$-modules V including the trivial $\mathbb{R}[K]$-module $\mathbb{R}$. We then extend the first and third results to topological spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.

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arXiv2022

Bounded cohomology and homotopy colimits

The comparison map from bounded cohomology to singular cohomology plays an important role in the study of bounded cohomology theory and its applications. The vanishing and covering theorems of Gromov and Ivanov show interesting and useful properties of the comparison map under appropriate assumptions. We discuss an approach to these theorems, based on the general homotopy-theoretic properties of the comparison map, and obtain new proofs and refined versions of these results.

math.AT↗
arXiv2022

Flat functors in higher topos theory

For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we study necessary and sufficient conditions for a functor $f \colon \mathscr{C} \to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the $n$-topos $\mathcal{P}(\mathscr{C})_n$ of presheaves on $\mathscr{C}$ for any $n \geq 1$. These results generalize and unify results of Lurie for $n=\infty$ and classical characterizations of flat functors (Diaconescu's theorem) for $n=1$. Interestingly, for $n=\infty$, our analogue of Diaconescu's theorem requires hypercompleteness. As an application, we show that the $\infty$-topos associated to an $n$-site behaves as an $n$-localic $\infty$-topos with respect to hypercomplete $\infty$-topoi.

math.CT↗
arXiv2022

Higher weak (co)limits, adjoint functor theorems, and higher Brown representability

We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\infty$-categories for any $n \geq 1$.

math.CT↗
arXiv2022

Dévissage for Waldhausen K-theory

A dévissage-type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $\mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $\mathscr{C}$ when a dévissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between dévissage theorems of single type and of multiple type, depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original dévissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of dévissage-type theorems and prove a general dévissage theorem of single type and a general dévissage theorem of multiple type.

math.KT↗
arXiv2022

Higher homotopy categories, higher derivators, and K-theory

For every $\infty$-category $\mathscr{C}$, there is a homotopy $n$-category $\mathrm{h}_n \mathscr{C}$ and a canonical functor $γ_n \colon \mathscr{C} \to \mathrm{h}_n \mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy $n$-categories, we introduce the notion of an $n$-derivator and study the main examples arising from $\infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\infty$-derivators and prove that the canonical comparison map from the Waldhausen $K$-theory of $\mathscr{C}$ to the $K$-theory of the associated $n$-derivator $\mathbb{D}_{\mathscr{C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator $K$-theory of $\infty$-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy $n$-category, we also define a $K$-theory space $K(\mathrm{h}_n \mathscr{C}, \mathrm{can})$ associated to $\mathrm{h}_n \mathscr{C}$. We prove that the canonical comparison map from the Waldhausen $K$-theory of $\mathscr{C}$ to $K(\mathrm{h}_n \mathscr{C}, \mathrm{can})$ is $n$-connected.

math.KT↗
arXiv2022

The simplicial coalgebra of chains under three different notions of weak equivalence

We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring R. The weak equivalences are given by: (1) an R-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an R-homology equivalence between universal covers, and (3) R-homology equivalences. Analogously, for any field F, we construct three model structures on the category of connected simplicial cocommutative F-coalgebras. The weak equivalences in this context are (1') maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2') maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3') quasi-isomorphisms. Building on previous work of Goerss in the context of (3)-(3'), we prove that, when F is algebraically closed, the simplicial F-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each pair of model categories. More generally, when F is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of F.

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arXiv2022

On the simplicial volume and the Euler characteristic of (aspherical) manifolds

A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov's question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.

math.AT↗
arXiv2020

On transfer maps in the algebraic $K$-theory of spaces

We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to \emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.

math.KT↗
arXiv2020

Topological manifold bundles and the $A$-theory assembly map

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through the assembly map of $A$-theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended $A$-theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended $A$-theory characteristic becomes highly connected as the manifold dimension increases.

math.AT↗
arXiv2019

On the $h$-cobordism category. I

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

math.AT↗
arXiv2019

Adjoint functor theorems for $\infty$-categories

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

math.CT↗
arXiv2018

A cobordism model for Waldhausen $K$-theory

We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's $S_{\bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.

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