Four approaches to cohomology theories with reality

  title={Four approaches to cohomology theories with reality},
  author={J. P. C. Greenlees},
  journal={arXiv: Algebraic Topology},
  • J. Greenlees
  • Published 25 May 2017
  • Mathematics
  • arXiv: Algebraic Topology
We give an account of well known calculations of the RO(Q)-graded coefficient rings of some of the most basic Q-equivariant cohomology theories, where Q is a group of order 2. One purpose is to advertise the effectiveness of the Tate square, showing it has advantages over the slice spectral sequences in algebraically simple cases. A second purpose is to give a single account showing how to translate between the languages of different approaches. [v2 corrects some typos and adds some thanks and… 

Figures from this paper

The representation-ring-graded local cohomology spectral sequence for BPℝ⟨3⟩
Abstract The purpose of this paper is to examine the calculational consequences of the duality proved by the first author and Meier in [13], making them explicit in one more example. We give an
Picard groups and duality for real Morava E–theories
We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal
Bredon motivic cohomology of the complex numbers
Over the complex numbers, we compute the C2-equivariant Bredon motivic cohomology ring with Z/2 coefficients. By rigidity, this extends Suslin’s calculation of the motivic cohomology ring of
On the $RO(Q)$-graded coefficients of Eilenberg-MacLane spectra
Let& denote the cyclic group of order two. Using the Tate diagram we compute the'$(&)-graded coefficients of Eilenberg-MacLane &-spectra and describe their structure as a module over the coefficients
Conjugation spaces are cohomologically pure
Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees
Commuting matrices and Atiyah's real K‐theory
We describe the C2 ‐equivariant homotopy type of the space of commuting n ‐tuples in the stable unitary group in terms of Real K‐theory. The result is used to give a complete calculation of the
Bredon cohomology of finite dimensional $C_p$-spaces
For finite dimensional free $C_p$-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of $S^0$ is used to prove the non-existence of certain $C_p$-maps. These are
The cohomology of C2-equivariant 𝒜(1) and the homotopy of koC2
We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to
The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture
In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We
Equivariant Gorenstein Duality
This thesis concerns the study of two flavours of duality that appear in stable homotopy theory and their equivariant reformulations. Concretely, we look at the Gorenstein duality framework


On the nonexistence of elements of Kervaire invariant one
We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions
  • N. Ricka
  • Mathematics
    Glasgow Mathematical Journal
  • 2015
Abstract We show that the $\mathbb{Z}$ /2-equivariant nth integral Morava K-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal
The C2–spectrum Tmf1(3) and its invertible modules
We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In particular, we compute their $C_2$-equivariant Picard groups and the $C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies
An Atiyah-Hirzebruch spectral sequence for KR-theory
We construct a spectral sequence for computing KR-theory, analogous to the spectral sequence relating motivic cohomology to algebraic K-theory.
Operations in equivariant [Zopf]/ p -cohomology
Equivariant stable homotopy and Segal's Burnside ring conjecture
Analogous results were proved later for KO, in the generality of compact Lie groups, by Atiyah and Segal [8], and for KFq, the algebraic K-theory spectrum associated to the finite field Fq, by Rector
M.J.Hopkins and D.C.Ravenel “The slice spectral sequence for the C4 analog of real K-theory
  • Forum Math
  • 2017