Eric B. Kahn
We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group.
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We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Hay…
We compute the $RO(A)$-graded coefficients of $A$-equivariant complex and real topological $K$-theory for $A$ a finite elementary abelian $2$-group, together with all products, transfers, restrictions, power operations, and Adams operations.
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.
The structure of the BP<n>-cohomology of elementary abelian p-groups is studied, obtaining a presentation expressed in terms of BP-cohomology and mod-p singular cohomology, using the Milnor derivations. The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra.
Let $G$ be a compact Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk--Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups.
The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU-cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case. These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Gre…
A general notion of detection is introduced and used in the study of the cohomology of elementary abelian 2-groups with respect to the spectra in the Postnikov tower of orthogonal K-theory. This recovers and extends results of Bruner and Greenlees and is related to calculations of the (co)homology of the spaces of the associated Omega-spectra by Stong and by Cowen Morton.
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.
In this paper, we determine the connective K-cohomology with reality of elementary abelian $2$-groups as a module over $\mathbb{Z}[v_1,a]$, where $v_1$ is the equivariant Bott class and $a$ the Euler class of the sign representation. This gives in particular a new approach to the computation of the connective real K-theory of such groups. The originality here is to make all computations in the $\mathbb{Z}/2$-equivariant stable category, considering only $\mathbb{Z}/2$-equivariant cohomology theories, and to use relative homological algebra over certain subalgebras of the equivariant Steenrod a…
Representation-graded Bredon homology of elementary abelian 2-groups
本文计算了所有初等阿贝尔2-群在常数模2 Mackey函子系数下的表示分阶Bredon同调环，给出了基于pre-Euler类和逆Thom类的极小展示，并由此推导出几何不动点环与局部化扭曲上同调环的计算结果。
We determine the BP-module structure, mod higher filtration, of the main part of the BP-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.
We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a sa…
We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $σ\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.
We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem is to determine the set of $\mathcal {A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each $k \geq 0$, the set of {\it $k$ partially $\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i…
In this paper we study E^*BV_k, where E=BP<m,n> is a cohomology theory with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n] (if m=0). We use ideas from the theory of multiple level structures, developed in earlier work of the author with John Greenlees. Our results apply when k is less than or equal to w=n+1-m. If k<w we find that E^*BV_k has no v_m-torsion. When k=w, we show that the v_m-torsion is annihilated by the ideal I_{n+1}=(v_m,...,v_n), and that it is a free module on one generator over the ring F_p[[x_0,...,x_{w-1}]]. We give three very different formulae for this g…
Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for $G=(\mathbb{Z}/2)^n$ was completely calculated by Bruner and Greenlees. In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to $p>2$ prime. We also identify the resulting spectra, which are products of Eilenberg-Mac Lane spectra, and finitely many finite Postnikov towers. For $p=2$, we also reconcile our answer completely with the result of Bruner and Greenlees, which is in a different form, and hence the comparison involves some non-trivial combinatorics…
Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
本文证明 Hill 特征函数满射到内部算子且纤维唯一极大元恰为饱和转移系统，并通过模格上的火柴棒博弈和有界偏序集递归，确定了秩二初等阿贝尔群转移系统的完整格结构。
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, I correct errors in my paper (Kodai Math. J. (2015)) about gamma filtrations for classifying spaces for abelian p-groups which are not elementary. We also extend Chetard's results for such 2-groups to p-groups for odd prime.
The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group $\mathcal{M}_σ$ of a compact closed surface $X$ of genus $σ$ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumeration of finite abelian subgroups.
We use topological methods to compute the mod p cohomology of certain p-groups. More precisely we look at central Frattini extensions of elementary abelian by elementary abelian groups such that their defining k-invariants span the entire image of the Bockstein. We show that if p is sufficiently large, then the mod p cohomology of the extension can be explicitly computed as an algebra.
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