[Paper Review] The $\mathrm{RO}(G)$-Graded Cohomology of the Equivariant Classifying Space $B_G\mathrm{SU}(2)$

Zev Chonoles|arXiv (Cornell University)|Aug 2, 2018

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper computes the RO(C₂)-graded Bredon cohomology of the equivariant classifying space BC₂SU(2) with coefficients in the Burnside ring Mackey functor. Using equivariant even-dimensional freeness and multiplicative comparison theorems, it establishes that the cohomology is generated by two elements—c in dimension 4σ and C in dimension 4+4σ—satisfying the relation c² = ε⁴c + ξ²C, where ε and ξ are elements of the cohomology of a point.

ABSTRACT

We compute the additive structure of the $\mathrm{RO}(C_n)$-graded Bredon equivariant cohomology of the equivariant classifying space $B_{C_n}\mathrm{SU}(2)$, for any $n$ that is either prime or a product of distinct odd primes, and we also compute its multiplicative structure for $n=2$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of $B_{C_2}\mathrm{SU}(2)$ is generated by two elements subject to a single relation: writing $σ$ for the sign representation of $C_2$ in $\mathrm{RO}(C_2)$, the generators are an element $c$ in dimension $4σ$ and an element $C$ in dimension $4+4σ$, satisfying the relation $c^2 = ε^4 c + ξ^2 C$, where $ε$ and $ξ$ are elements of the cohomology of a point. Throughout, we take coefficients in the Burnside ring Mackey functor $A$. The key tools used are equivariant "even-dimensional freeness" and "multiplicative comparison" theorems for $G$-cell complexes, both proven by Lewis in [Lew88] and subsequently refined by Shulman in [Shu10], and with the former theorem extended by Basu and Ghosh in [BG16]. The latter theorem enables us to compute the multiplicative structure of the cohomology of $B_{C_2}\mathrm{SU}(2)$ by embedding it in a direct sum of cohomology rings whose structure is more easily understood. Both theorems require the cells of the $G$-cell complex to be attached in a well-behaved order, and a significant step in our work is to give $B_{C_n}\mathrm{SU}(2)$ a satisfactory $C_n$-cell complex structure.

Motivation & Objective

• To determine the additive and multiplicative structure of RO(G)-graded Bredon cohomology for the equivariant classifying space BCₙSU(2), where G = Cₙ and n is prime or a product of distinct odd primes.
• To construct a well-behaved Cₙ-cell complex structure on BCₙSU(2) to enable application of equivariant cohomology theorems.
• To compute the multiplicative structure of eH∗C₂(BC₂SU(2)+; A) as an algebra over eH∗C₂(S⁰; A), identifying generators and relations.
• To extend and apply the even-dimensional freeness and multiplicative comparison theorems of Lewis and Shulman to equivariant cohomology computations.
• To verify the algebraic relation between generators using restriction and transfer maps to the fixed-point subspaces.

Proposed method

• Constructs a Cₙ-cell complex structure on BCₙSU(2) using equivariant projective spaces of Cₙ-representations.
• Applies the even-dimensional freeness theorem (generalized by Basu and Ghosh) to deduce that the cohomology of BCₙSU(2) is free over the cohomology of a point in even RO(G)-degrees.
• Uses the multiplicative comparison theorem (Shulman, after Lewis) to embed the cohomology of BC₂SU(2) into a direct sum of simpler cohomology rings.
• Employs restriction and transfer maps via the Borel construction and cofiber sequences to analyze the cohomology of filtration stages.
• Identifies generators c and C in dimensions 4σ and 4+4σ using explicit constructions from equivariant spheres and projective spaces.
• Verifies the relation c² = ε⁴c + ξ²C by checking its image under restriction to fixed points and via the comparison theorem.

Experimental results

Research questions

• RQ1 What is the additive structure of RO(Cₙ)-graded Bredon cohomology of BCₙSU(2) for n prime or a product of distinct odd primes?
• RQ2 What is the multiplicative structure of eH∗C₂(BC₂SU(2)+; A) as an algebra over eH∗C₂(S⁰; A)?
• RQ3 How can the multiplicative structure be computed using comparison theorems and cell complex filtrations?
• RQ4 What is the algebraic relation between the generators of the cohomology ring in the C₂-case?
• RQ5 How do restriction and transfer maps help verify the cohomology relation c² = ε⁴c + ξ²C?

Key findings

• The RO(Cₙ)-graded Bredon cohomology of BCₙSU(2) with coefficients in the Burnside ring Mackey functor A is free as a module over eH∗Cₙ(S⁰; A) in even RO(Cₙ)-degrees.
• For G = C₂, the cohomology eH∗C₂(BC₂SU(2)+; A) is generated as an algebra by two elements: c in dimension 4σ and C in dimension 4+4σ.
• The generators satisfy the relation c² = ε⁴c + ξ²C, where ε and ξ are elements of eH∗C₂(S⁰; A), with ε in degree 0 and ξ in degree 2.
• The restriction of c to the fixed point is x₀, and the restriction of C is x₀(ε⁴ + ξ²x₀), consistent with the relation.
• The cohomology of the filtration stages PH(Wₙ)+ is isomorphic to eH∗C₂(PH(Wₙ₋₁)+) ⊕ Σ^{wₙ} eH∗C₂(S⁰; A), with wₙ = 4n.
• The natural map eH∗C₂(BC₂SU(2)+; A) → lim←− eH∗C₂(PH(Wₙ)+; A) is an isomorphism, confirming the colimit structure of the cohomology.