Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the $RO(G)$-graded coefficients of dihedral equivariant cohomology
89
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We completely calculate the $RO(G)$-graded coefficients of ordinary equivariant cohomology where $G$ is the dihedral group of order $2p$ for a prime $p>2$ both with constant and Burnside ring coefficients. The authors first proved it for $p=3$ and then the second author generalized it to arbitrary $p$. These are the first such calculations for a non-abelian group.
rate research
Read More
This thesis consists of two main parts. In the second part, we recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences for RO(G)-graded Bredon homology and cohomology. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology of the equivariant classifying space B_Cp O(2). However, to complete this sample computation, we need to know the cohomology of complex projective space. This calculation was done in a 1988 paper by Gaunce Lewis, but relies on a theorem whose proof as given was incorrect. We spend the first part of this thesis providing a correct proof and summarizing the results of Lewiss paper.
This note contains a generalization to $p>2$ of the authors previous calculations of the coefficients of $(mathbb{Z}/2)^n$-equivariant ordinary cohomology with coefficients in the constant $mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum $Phi^{(mathbb{Z}/p)^n}Hmathbb{Z}/p$, and more generally, the $mathbb{Z}$-graded coefficients of the localization of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$ by inverting any chosen set of embeddings $S^0rightarrow S^{alpha_i}$ where $alpha_i$ are non-trivial irreducible representations. We also calculate the $RO(G)^+$-graded coefficients of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$, which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the non-derived part, which has a nice algebraic description.)
This survey paper describes two geometric representations of the permutation group using the tools of toric topology. These actions are extremely useful for computational problems in Schubert calculus. The (torus) equivariant cohomology of the flag variety is constructed using the combinatorial description of Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation representations on equivariant and ordinary cohomology are identified in terms of irreducible representations of the permutation group. We show how to use the permutation actions to construct divided difference operators and to give formulas for some localizations of certain equivariant classes. This paper includes several new results, in particular a new proof of the Chevalley-Monk formula and a proof that one of the natural permutation representations on the equivariant cohomology of the flag variety is the regular representation. Many examples, exercises, and open questions are provided.
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
We give an algorithm for calculating the RO(S^1)-graded TR-groups of F_p, completing the calculation started by the second author. We also calculate the RO(S^1)-graded TR-groups of Z with mod p coefficients and of the Adams summand ell of connective complex K-theory with V(1)-coefficients. Some of these calculations are used elsewhere to compute the algebraic K-theory of certain Z-algebras.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
