Local cohomology techniques in equivariant homotopy theory, introduced by John Greenlees, may be applied to understand homology of classifying spaces through other equivariant data. In this paper we relate the local cohomology filtration to the families filtration. By doing so, we may identify geometry codified by the local cohomology filtration in the setting of equivariant bordism. The constructions which arise are naturally analyzed by localized K-theory machinery due to Atiyah and Segal, which we review.
We give a description of real equivariant bordism for elementary abelian 2-groups, which is similar to the description of complex equivariant bordism for the group S^1 x ... x S^1 given by Hanke in 2005.
We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.
In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $Ω^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques the author developed in the complex setting. Because we are working in the real setting only with Z/2, these techniques simplify greatly.
We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is known to hold and we highlight the properties that permit one to prove the conjecture in these cases.
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three…
By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. This is used in order to determine the image of the Pontrjagin-Thom map for toral G.
The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of …
For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian 2-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A = C_2$ due to Hanke-Wiemeler (2018). We work in the framework of global homotop…
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that our construction recovers the construction of Wasserman, providing a new perspective on equivariant bordism. We connect the results with bordisms of global quotient orbifolds, utilizing the machinery of Gepner-Henriques to describe bordisms of framed orbifolds in terms of equivaria…
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented …
A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the $\mathbb{Z}/2$-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). Our proof combines a computation of the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigati…
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.
In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
Homological Description of Equivariant Geometric Bordism
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；College of Mathematics and Sciences, Shanghai Normal University（上海师范大学数理学院）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
Arone and Lesh constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg-MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and of another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh's description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups…
Let $G/H$ be a homogeneous variety, and let $X$ be a $G$-equivariant embedding of $G/H$such that the number of $G$-orbits in $X$ is finite. We show that the equivariant Borel-Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel-Moore homologies of the $G$-orbits. If $T$ is a maximal torus of $G$ such that each $G$-orbit has a $T$-fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel-Moore homology of $X$. We apply our findings to certain wonderful compactifications as well as to double flag varie…
We identify the equivariant structure of the filtered pieces of the motivic filtration defined by Bhatt, Morrow and Scholze on the topological Hochschild cohomology spectrum of polynomial algebras over $\mathbb{F}_p$.
We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice dimension of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg-Mac Lane spectrum associated to a Mackey functor for a cyclic $p$-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullba…
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 produ…
We study the Duflot filtration on the Borel equivariant cohomology of smooth manifolds with a smooth $p$-torus action. We axiomatize the filtration and prove analog of several structural results about equivariant cohomology rings in this setting. We apply this abstract theory to study the $\mathbb{F}_p$ cohomology rings of classifying spaces of compact Lie groups, and show how to recover geometric results about the cohomology of $BG$ using equivariant cohomology. This includes some results about detection on subgroups and restrictions on associated primes that were previously only known for fi…
This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a $G$-spectrum is slice $n$-connective. In particular, we show that a $G$-spectrum is slice greater than or equal to $n$ if and only if for all subgroups $H$, the $H$-geometric fixed points are $(n/|H|-1)$-connected. We use this to determine when smashing with a virtual representation sphere $S^V$ induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary $C_p$-spectrum and show how a ve…
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.
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 h…
Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor $\underline{\mathbb{F}}_p$, denoted $H\underline{\mathbb{F}}_p$, is modeled by the free $\mathbb{F}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length' filtration $\{(H\underline{\mathbb{F}}_p)_n\}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H\underline{\mathbb{F}}_p)_{p^k}/(H\underline{\mathbb{F}}_p)_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p…
Let $k$ be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over $Spec(k)$ with the $C_2$-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of $MGL$ and $MR$, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of $MR$ are e…
Let $G$ be a elementary abelian $2$-group and $X$ be a manifold with a locally standard action of $G$. We provide a criterion to determine the syzygy order of the $G$-equivariant cohomology of $X$ with coefficients over a field of characteristic two using a complex associated to the cohomology of the face filtration of the manifold with corners $X/G$. This result is the real version of the quotient criterion for locally standard torus actions developed by M. Franz in arXiv:1205.4462.
We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.
We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We al…
Let X be a "nice" space with an action of a torus T. We consider the Atiyah-Bredon sequence of equivariant cohomology modules arising from the filtration of X by orbit dimension. We show that a front piece of this sequence is exact if and only if the H^*(BT)-module H_T^*(X) is a certain syzygy. Moreover, we express the cohomology of that sequence as an Ext module involving a suitably defined equivariant homology of X. One consequence is that the GKM method for computing equivariant cohomology applies to a Poincare duality space if and only if the equivariant Poincare pairing is perfect.
The slice filtration is a filtration of equivariant spectra. While the tower is analogous to the Postnikov tower in the nonequivariant setting, complete slice towers are known for relatively few $G$-spectra. In this paper, we determine the slice tower for all $G$-spectra of the form $S^n \wedge H\underline{\mathbb{Z}}$ where $n\geq 0$ and $G$ is a cyclic $p$-group for $p$ an odd prime.
An old result of the first author and David Lieberman says that if a compact Kaehler manifold X admits a holomorphic vector field V having at least one zero, then the Dolbeault cohomology algebra H^*(X, Ω^*) of X is isomorphic with the bigraded algebra associated to a certain filtered graded ring. If V has isolated zeros this is the graded algebra associated to a certain filtration of the coordinate ring of the scheme Z defined by zero(V). The purpose of this note is to show that if V is generated by a torus action then equivariant Dolbeault cohomology can be used to give direct geometric proo…
Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the "topological complex", is defined using the orbit filtration of $X$. The second, which we call the "algebraic complex", is defined just in terms of the unstable $\mathrm{A}$-module $\mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of un…
We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates…
