arXiv subjects

Stefan Schwede

Publications and source records attributed to Stefan Schwede.

At least 19 recordsLinked to original sources

arXiv2026

A Real-global equivariant Segal--Becker splitting, explicit Brauer induction, and global Adams operations

We prove a splitting result in global equivariant homotopy theory that is a simultaneous refinement of the Segal--Becker splitting and its `Real' and equivariant generalizations, and of the explicit Brauer induction of Boltje and Symonds. We show that the morphism of ultra-commutative Real-global ring spectra from $Σ^\infty_+ B_{\text{gl}}U(1)$ to the Real-global K-theory spectrum that classifies the tautological Real $U(1)$-representation admits a section on underlying Real-global infinite loop spaces. We prove that this global Segal--Becker splitting induces the classical Segal--Becker splittings on equivariant cohomology theories, and that it induces the Boltje--Symonds explicit Brauer induction on equivariant homotopy groups. As an application we rigidify the unstable Adams operations in Real-equivariant K-theory to global self-maps of the Real-global space $\mathbf{BUP}$.

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arXiv2024

Character theory and Euler characteristic for orbispaces and infinite groups

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 $Γ_\frac{p-1}{2}$ 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 $K$.

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arXiv2024

Representation-graded Bredon homology of elementary abelian 2-groups

We calculate the representation-graded Bredon homology rings of all elementary abelian 2-groups with coefficients in the constant mod-2 Mackey functor. We exhibit minimal presentations for these rings as quotients of the polynomial algebra on the pre-Euler and inverse Thom classes of all nontrivial characters, subject to an explicit finite list of relations arising from orientability properties. Two corollaries of our presentation are the calculation, originally due to Holler and Kriz, of the geometric fixed point rings, and a strengthening of a calculation of Balmer and Gallauer of the localized twisted cohomology ring.

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arXiv2024

The universal property of bordism of commuting involutions

We propose a formalism to capture the structure of the equivariant bordism rings of smooth manifolds with commuting involutions. We introduce the concept of an oriented el$_2^{RO}$-algebra, an algebraic structure featuring representation graded rings for all elementary abelian 2-groups, connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom class; this data is subject to one exactness property. Besides equivariant bordism, oriented global ring spectra also give rise to oriented el$_2^{RO}$-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el$_2^{RO}$-algebras are delocalized generalizations of global 2-torsion group laws. Our main result shows that equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra. Several other interesting equivariant homology theories can also be characterized, on elementary abelian 2-groups, by similar universal properties. We prove that stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el$_2^{RO}$-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el$_2^{RO}$-algebra with an orientation that is both additive and invertible.

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arXiv2023

Chern classes in equivariant bordism

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner-Floyd-Chern classes in the $MU$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees-May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $MU$-completion theorem of Greenlees-May and La Vecchia.

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arXiv2022

Global stable splittings of Stiefel manifolds

We prove global equivariant refinements of Miller's stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.

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arXiv2022

Global algebraic K-theory

We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $Ω$-spectrum that keeps track of genuine $G$-equivariant infinite loop spaces, for all finite groups $G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or `Swan K-theory' into one highly structured object.

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arXiv2022

Splittings of global Mackey functors and regularity of equivariant Euler classes

We establish natural splittings for the values of global Mackey functors at orthogonal, unitary and symplectic groups. In particular, the restriction homomorphisms between the orthogonal, unitary and symplectic groups of adjacent dimensions are naturally split epimorphisms. The interest in the splitting comes from equivariant stable homotopy theory. The equivariant stable homotopy groups of every global spectrum form a global Mackey functor, so the splittings imply that certain long exact homotopy group sequences separate into short exact sequences. For the real and complex global Thom spectra $\mathbf{MO}$ and $\mathbf{MU}$, the splittings imply the regularity of various Euler classes related to the tautological representations of $O(n)$ and $U(n)$.

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arXiv2020

Proper equivariant stable homotopy theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective `genuine' indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an $RO(G)$-grading. Our model for genuine proper $G$-equivariant stable homotopy theory is the category of orthogonal $G$-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of $G$. This class of $π_*$-isomorphisms is part of a symmetric monoidal stable model structure and the associated tensor triangulated homotopy category is compactly generated. Every orthogonal $G$-spectrum represents an equivariant cohomology theory on the category of $G$-spaces, depending only on the `proper $G$-homotopy type', tested by fixed points under all compact subgroups. An important special case are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties, in the sense of geometric group theory; for example, the $G$-sphere spectrum is a compact object in the equivariant homotopy category if the universal space for proper $G$-actions has a finite $G$-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper $G$-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by $G$-vector bundles. Via this description, we can identify the previously defined $G$-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal $G$-spectra.

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arXiv2020

Homotopy invariance of convolution products

The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $\infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.

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arXiv2019

Categories and orbispaces

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered. A global equivalence is a functor between small categories that induces weak equivalences of nerves of the categories of $G$-objects, for all finite groups $G$. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equivalent to the homotopy theory of orbispaces in the sense of Gepner and Henriques. Every cofibrant category in this global model structure is opposite to a complex of groups in the sense of Haefliger.

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arXiv2019

Orbispaces, orthogonal spaces, and the universal compact Lie group

This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of `spaces with an action of the universal compact Lie group'. The upshot is a novel way to construct and study genuine cohomology theories on stacks, orbifolds, and orbispaces, defined from stable global homotopy types represented by orthogonal spectra. The universal compact Lie group (which is neither compact nor a Lie group) is a well known object, namely the topological monoid $\mathcal L$ of linear isometric self-embeddings of $\mathbb R^\infty$. The underlying space of $\mathcal L$ is contractible, and the homotopy theory of $\mathcal L$-spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces. However, the monoid $\mathcal L$ contains copies of all compact Lie groups in a specific way, and we define global equivalences of $\mathcal L$-spaces by testing on corresponding fixed points. We establish a global model structure on the category of $\mathcal L$-spaces and prove it to be Quillen equivalent to the global model category of orthogonal spaces, and to the category of orbispaces, i.e., presheaves of spaces on the global orbit category.

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arXiv2018

Global homotopy theory

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global homotopy theory, modeled by orthogonal spaces respectively orthogonal spectra under global equivalences. Specific topics include the global stable homotopy category, operations on equivariant homotopy groups, global model structures, and ultra-commutative multiplications. The book includes many explicit examples and detailed calculations.

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arXiv2017

Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case, for example the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory, i.e., we study the simultaneous behavior for all compact Lie groups at once.

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arXiv2012

Topological triangulated categories

In this paper we explain certain systematic differences between algebraic and topological triangulated categories. A triangulated category is algebraic if it admits a differential graded model, and topological if it admits a model in the form of a stable cofibration category. The precise statements use the 'n-order' of a triangulated category, for a natural number n. The n-order is a non-negative integer (or infinity) and measures `how strongly' n annihilates objects of the form Y/n. We show the following results: the n-order of an algebraic triangulated category is infinite; for every prime p, the p-order of a topological triangulated category is at least p-1; the p-order of the p-local stable homotopy category is exactly p-1. In particular, the p-local stable homotopy category is not algebraic for any prime p. As a tool we develop certain foundations about enrichments of cofibration categories by Delta-sets; in particular we generalize the theory of `framings' (or `cosimplicial resolutions') from model categories to cofibration categories.

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arXiv2008

Algebraic versus topological triangulated categories

The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the source categories are usually very `non-additive' before passing to homotopy classes of morphisms. Because of their origin I refer to these examples as `topological triangulated categories'. In these extended talk notes I explain some systematic differences between these two kinds of triangulated categories. There are certain properties -- defined entirely in terms of the triangulated structure -- which hold in all algebraic examples, but which fail in some topological ones. These differences are all torsion phenomena, and rationally there is no difference between algebraic and topological triangulated categories.

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arXiv2007

Triangulated categories without models

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.

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arXiv2006

On the homotopy groups of symmetric spectra

The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model structure. About the only disadvantage of this model is that the stable equivalences cannot be defined by inverting those morphisms which induce isomorphisms on homotopy groups, because this would leave too many homotopy types. In this sense the naively defined homotopy groups are often `wrong`, and then their precise relationship to the `true` homotopy groups (i.e., morphisms in the stable homotopy category from sphere spectra) appears mysterious. In this paper I discuss and exploit extra algebraic structure on the naively defined homotopy groups of symmetric spectra, namely a special kind of action of the monoid of injective self maps of the natural numbers. This extra structure clarifies several issues about homotopy groups and stable equivalences and explains why the naive homotopy groups are not so wrong after all. For example, the monoid action allows a simple characterization of semistable symmetric spectra.

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