Mackey Functors Research Articles

Mini-Workshop: Homological aspects of TDLC-groups

This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific topics include finiteness conditions for locally compact groups, Mackey functors and Bredon cohomology for topological groups, connections to condensed mathematics, connections to \ell^{2} -invariants and \Sigma -invariants.

Green correspondence on centric Mackey functors over fusion systems

In this paper we give a definition of (centric) Mackey functor over a fusion system (Definitions 2.26 and 2.29) which generalizes the notion of Mackey functor over a group. In this context we prove that, given some conditions on a related ring, the centric Burnside ring over a fusion system (as defined in [1]) acts on any centric Mackey functor (Proposition 2.43). We also prove that the Green correspondence holds for centric Mackey functors over fusion systems (Theorem 4.37). As a mean to prove this we introduce a notion of relative projectivity for centric Mackey functors over fusion systems (Definition 3.1) and provide a decomposition of a particular product in O(Fc)⊔ (Definition 2.12) in terms of the product in O((NF)c)⊔ (Theorem 4.27).

On curves in K-theory and TR

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line \mathbf{S}[t] as a functor defined on the \infty -category of cyclotomic spectra with values in the \infty -category of spectra with Frobenius lifts, refining a result of Blumberg–Mandell. We define the notion of an integral topological Cartier module using Barwick’s formalism of spectral Mackey functors on orbital \infty -categories, extending the work of Antieau–Nikolaus in the p -typical setting. As an application, we show that TR evaluated on a connective \mathbf{E}_1 -ring admits a description in terms of the spectrum of curves on algebraic K-theory, generalizing the work of Hesselholt and Betley–Schlichtkrull.

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Mackey Profunctors

In the standard theory of Mackey functors for a group G G , the group is either finite, or a compact Lie group. In the present paper, we develop an alternative notion of a Mackey profunctor that works better for profinite groups. The most important practical example is G = Z ^ G=\widehat {\mathbb {Z}} , the profinite completion of the group of the integers, and in this case, our Mackey profunctors appear naturally in the theory of cyclotomic traces.

On the RO(Q)-graded coefficients of Eilenberg–MacLane spectra

Let Q denote the cyclic group of order two. Using the Tate diagram we compute the RO(Q)-graded coefficients of Eilenberg–MacLane Q-spectra and describe their structure as modules over the coefficients of the Eilenberg–MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also obtain the multiplicative structure on the RO(Q)-graded coefficients.

The equivalence between rational G-sheaves and rational G-Mackey functors for profinite G

AbstractFor G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed.This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis.

COHOMOLOGICAL MACKEY 2-FUNCTORS

AbstractWe show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

Homotopy Mackey functors of equivariant algebraic K-theory

Given a finite group G acting on a ring R, Merling constructed an equivariant algebraic K-theory G-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide an interpretation of the homotopy Mackey functors of equivariant algebraic K-theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the nth algebraic K-groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along homomorphisms of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic K-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K-theory of endomorphism rings, the K-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings.

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 $U(n)$.

The equivariant parametrized h-cobordism theorem, the non-manifold part

We construct a map from the suspension G-spectrum ΣG∞M of a smooth compact G-manifold to the equivariant A-theory spectrum AG(M), and we show that its fiber is, on fixed points, a wedge of stable h-cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible with tom Dieck style splitting formulas on fixed points. In order to reconcile different definitions of the suspension G-spectrum as a spectral Mackey functor, we present a new perspective on spectral Mackey functors, viewing them as multifunctors on indexing categories for “rings on many objects” and modules over such. This perspective should be of independent interest.

The C2-equivariant cohomology of complex projective spaces

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of C2, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the RO(C2)-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bézout's theorem.

Free Incomplete Tambara Functors are Almost Never Flat

AbstractFree algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.

On the RO(G)-graded coefficients of Q8 equivariant cohomology

We calculate the RO(G)-graded coefficients of HZ_, the Eilenberg-MacLane spectrum of constant Mackey functor for quaternion group Q8.

The spectrum of derived Mackey functors

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by “ungluing” closed points. In order to compute the spectrum, we provide a new description of Kaledin’s category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.

Invertible K(2)–local E–modules in C4–spectra

We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order 32 generated by $ E\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\wedge S^{7+\sigma}$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum $E$.

The freeness theorem for equivariant cohomology of Rep(C2)-complexes

Let C2 be the cyclic group of order two. We show that the RO(C2)-graded Bredon cohomology of a finite Rep(C2)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor F2_. This paper corrects some errors in Kronholm's proof of this freeness theorem. It also extends the freeness result to finite type complexes, those with finitely many cells of each fixed-set dimension. We give a counterexample showing the theorem does not hold for locally finite complexes.

Real topological Hochschild homology

This paper interprets Hesselholt and Madsen’s real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for group algebras. Using these structural results we determine the homotopy type of THR( \mathbb F_p ) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of H \mathbb F_p . We then calculate the homotopy type of THR( \mathbb Z ) away from the prime 2, and the homotopy ring of its geometric fixed-points spectrum \Phi^{\mathbb Z/2} THR( \mathbb Z ).

A structure theorem for RO(C2)–graded Bredon cohomology

Let $C_2$ be the cyclic group of order two. We present a structure theorem for the $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces using coefficients in the constant Mackey functor $\underline{\mathbb{F}_2}.$ We show that, as a module over the cohomology of the point, the $RO(C_2)$-graded cohomology of a finite $C_2$-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. The shifts are by elements of $RO(C_2)$ corresponding to actual (i.e. non-virtual) $C_2$-representations. This decomposition lifts to a splitting of genuine $C_2$-spectra.

Idempotent characters and equivariantly multiplicative splittings of K‐theory

We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude that the idempotent summands of $G$-equivariant topological $K$-theory and the corresponding summands of the $G$-equivariant sphere spectrum admit exactly the same flavors of equivariant commutative ring structures, made precise in terms of Hill-Hopkins-Ravenel norm maps. This paper is a sequel to the author's earlier work on multiplicative induction for the Burnside ring and the sphere spectrum, see arXiv:1802.01938.