$RO(C_p \times C_p)$-graded cohomology of universal spaces and the coefficient ring

This paper computes the $RO(C_p \times C_p)$-graded Bredon cohomology of equivariant universal and classifying spaces with constant $\underline{\mathbb{F}_p}$ coefficients, providing an explicit description of the resulting coefficient ring and applying these results to study lifts of cohomology operations via equivariant complex projective spaces.

The Gist

Imagine you are an architect trying to understand the structure of a massive, multi-dimensional building. But this isn't a normal building; it's a "symmetry building" where every room, hallway, and floor has a specific rule about how it can be rotated, flipped, or shifted without breaking.

In mathematics, this field is called Equivariant Homotopy Theory. The "building" is a space with a group acting on it (like rotating a square), and the "blueprints" are cohomology theories.

This paper by Surojit Ghosh and Ankit Kumar is like a team of master surveyors who have finally mapped out the foundation and the load-bearing walls of a very specific, complex type of symmetry building: one governed by the group $C_p \times C_p$. This group is like a grid of two independent rotating wheels (one for prime $p$, one for prime $p$ again).

Here is a breakdown of what they did, using simple analogies:

1. The Goal: Mapping the "Zero Point"

In math, before you can measure a complex shape, you need to know the properties of a single point. This is called the coefficient ring.

• The Analogy: Imagine trying to calculate the volume of a complex sculpture. First, you need to know the density and weight of the material itself.
• The Problem: For simple symmetries (like a single rotating wheel), mathematicians already knew the "density" of the material. But for this double-wheel symmetry ($C_p \times C_p$), the material was a mystery. It was too complex to measure directly.
• The Solution: The authors computed this "density" (the coefficient ring) explicitly. They figured out exactly how all the different "weights" (mathematical classes) multiply and interact. They built a complete dictionary of the rules for this specific symmetry.

2. The Tool: The "Tate Square"

To solve this, they used a powerful mathematical tool called the Tate Square.

• The Analogy: Imagine you want to see a statue in a dark room. You can't see it directly. So, you use a special mirror system (the Tate Square) that reflects the statue from three different angles.
  • One angle shows you the "free" parts (where the symmetry doesn't get stuck).
  • Another angle shows you the "fixed" parts (where the symmetry holds still).
  • The third angle combines them.
• The Work: The authors had to carefully analyze the "Universal Spaces" (the blueprints for these free parts) to make the mirror system work. They calculated the cohomology of these universal spaces first, which acted as the raw data needed to assemble the final picture of the coefficient ring.

3. The Result: A New "Periodic Table"

Once they had the coefficient ring, they essentially created a new periodic table for this symmetry group.

• The Discovery: They found that the structure is made of two types of blocks:
  1. Polynomial Blocks: These are the predictable, repeating patterns (like bricks in a wall).
  2. Inverted Blocks: These are the tricky parts, like "negative bricks" or holes in the wall that allow you to divide by certain numbers.
• The Breakthrough: They didn't just list the blocks; they wrote down the exact rules for how to stack them. For example, if you multiply Block A by Block B, you get Block C. This allows other mathematicians to build their own structures using these rules without having to reinvent the wheel.

4. The Application: The "Projective Space" Ladder

After mapping the foundation, they looked at a specific structure built on top of it: the Equivariant Complex Projective Space.

• The Analogy: Think of this as a ladder where each rung is a higher-dimensional version of the space.
• The Calculation: They figured out how to calculate the "weight" of a ladder made of $r$ rungs smashed together. They found a pattern: the weight of a big ladder is just a sum of shifted weights of smaller ladders. It's like realizing that a 10-story building's structural load is just a predictable combination of the loads of 1-story buildings.

5. The Final Twist: The "Unliftable" Operation

The paper ends with a surprising negative result about Cohomology Operations (which are like special tools you use to transform one shape into another).

• The Question: "If I have a tool that works on a simple 1-wheel symmetry, can I upgrade it to work on this complex 2-wheel symmetry?"
• The Answer: No.
• The Analogy: Imagine you have a wrench that fits a simple nut. You try to use it on a complex, double-nut mechanism. The authors proved that for certain specific sizes of nuts (degrees), the wrench simply cannot be upgraded. It's not just that it's hard; it's mathematically impossible. The structure of the complex symmetry is too rigid to allow the simple tool to pass through.

Summary

In short, Ghosh and Kumar:

1. Mapped the foundation of a complex symmetry group ($C_p \times C_p$) for the first time.
2. Built a dictionary of how all the mathematical pieces fit together.
3. Showed how to calculate the properties of complex shapes built on this foundation.
4. Proved a limitation: Some mathematical tools that work on simple symmetries simply cannot exist in this more complex world.

This work is a massive step forward because it gives mathematicians the "rules of the road" for a whole new class of symmetric spaces, allowing them to drive further into the landscape of equivariant homotopy theory without getting lost.

Technical Summary

Here is a detailed technical summary of the paper "RO($C_p \times C_p$)-Graded Cohomology of Universal Spaces and the Coefficient Ring" by Surojit Ghosh and Ankit Kumar.

1. Problem Statement

The central problem addressed in this work is the computation of the RO(G)-graded Bredon cohomology for the group $G = C_p \times C_p$ (the elementary abelian group of rank 2), where $p$ is a prime. Specifically, the authors aim to:

1. Determine the coefficient ring $\tilde{H}^\star_G(S^0; \mathbb{F}_p)$, which is the Bredon cohomology of a point with constant Mackey functor coefficients.
2. Compute the cohomology of equivariant universal spaces ($E C_p \mathbb{Z}/p$) and classifying spaces ($B C_p \mathbb{Z}/p$).
3. Analyze the module structure of the Bredon cohomology of smash powers of equivariant complex projective spaces.
4. Investigate the liftability of cohomology operations (specifically Steenrod operations) from subgroups to the full group $G$.

While the coefficient rings for cyclic groups ($C_p$) and the Klein four-group ($C_2 \times C_2$) were partially known, a complete description of the multiplicative structure for $C_p \times C_p$ (for both odd $p$ and $p=2$) remained an open challenge.

2. Methodology

The authors employ a combination of equivariant homotopy theory tools and algebraic computations:

• Tate Square Formalism: The primary computational engine is the Tate square (a homotopy pullback diagram). This relates the cohomology of a point to the cohomology of the universal space $E C_p \mathbb{Z}/p$ and its completion. The formula used is essentially:
  $$\tilde{H}^\star_G(S^0) \cong \text{fib}\left( \tilde{H}^\star_G(E C_p \mathbb{Z}/p_+) \to \tilde{H}^\star_G(E C_p \mathbb{Z}/p_+ \wedge F(E C_p \mathbb{Z}/p_+, H\mathbb{F}_p)) \right)$$
  This allows the authors to compute the coefficient ring by analyzing the map between the cohomology of the universal space and its function spectrum.

• Filtration of Universal Spaces: The universal space $E C_p \mathbb{Z}/p$ is constructed via a tower of cofiber sequences associated with families of subgroups. The authors analyze the Bredon cohomology of intermediate spaces $E F_i$ (associated with families of subgroups of order $p$) using spectral sequences (specifically the homotopy fixed point spectral sequence).

• Inductive Construction of Generators: For the odd prime case, the authors inductively construct new cohomology classes ($v_S, w_Q, z_Q$) associated with subsets of the subgroups of $G$. These classes are crucial for describing the polynomial part of the cohomology rings.

• Module Structure Analysis: To study the cohomology of smash powers of projective spaces ($P(U)^{\wedge r}$), the authors utilize the cofiber sequences defining projective spaces and prove vanishing theorems for specific gradings (Proposition 6.6), which implies that the boundary maps in the associated long exact sequences are trivial. This leads to a splitting of the cohomology.

• Lifting Obstruction: To address the liftability of operations, the authors examine the action of potential lifts on the cohomology of $P(U)^{\wedge r}$. They utilize the structure of Mackey functors to show that a non-trivial operation on the underlying non-equivariant space cannot extend to an equivariant operation due to the absence of specific summands in the target cohomology groups.

3. Key Contributions and Results

A. Cohomology of Universal Spaces ($E C_p \mathbb{Z}/p$)

• Odd Primes ($p$): The authors provide an explicit description of the polynomial part of $\tilde{H}^\star_G(E C_p \mathbb{Z}/p_+; \mathbb{F}_p)$. It is generated by classes $a_\beta, u_\beta, \kappa_\beta, a_{\alpha_\ell}, \kappa_{\alpha_\ell}$, and new classes $v_S, z_Q, w_Q$ indexed by subsets of the subgroups of $G$, subject to specific relations (Theorem 2.17).
• Case $p=2$ ($K_4$): They compute the full ring structure for $E C_2 \mathbb{Z}/2$, identifying an invertible class $v$ and describing the ring as a square-zero extension involving polynomial generators and additive generators (Theorem 3.7).

B. The Coefficient Ring ($\tilde{H}^\star_G(S^0; \mathbb{F}_p)$)

• Odd Primes: The polynomial part of the coefficient ring is determined (Theorem 4.17). It involves generators analogous to those in the universal space but with additional classes $k^{(i)}_{S_i}, \phi^{(i)}_{T_i}, \psi^{(i)}_{T_i}$ arising from the kernel of the boundary map in the Tate square.
• Case $p=2$ ($K_4$): The authors provide a complete, explicit description of the entire RO($K_4$)-graded ring, not just the polynomial part. This includes a detailed decomposition into polynomial and additive parts, along with a comprehensive list of relations (Theorem 5.19). This is a significant advancement over previous partial results for $K_4$.

C. Classifying Spaces ($B C_p \mathbb{Z}/p$)

• Using the projection map, the authors derive the RO($C_p$)-graded cohomology of the classifying space $B C_p \mathbb{Z}/p$ from the results on $E C_p \mathbb{Z}/p$. This yields explicit descriptions for both odd $p$ and $p=2$ (Corollary B and Proposition 3.8).

D. Equivariant Projective Spaces

• The paper establishes the additive structure of the Bredon cohomology for the $r$-fold smash power of the equivariant infinite complex projective space $P(U)^{\wedge r}$.
• The result shows that the cohomology splits as a direct sum of suspensions of the coefficient ring (Theorem 6.9). This generalizes known freeness results for cyclic groups to the $C_p \times C_p$ case.

E. Non-Liftability of Operations

• The authors prove a non-liftability theorem for cohomology operations. Specifically, they show that certain cohomology operations of degree $2r(p-1)$(for odd$p$) and$2r$(for$p=2$) that do not involve the Bockstein homomorphism **cannot be lifted** from the subgroup level to the$C_p \times C_p$ level (Theorem 6.14).
• This extends a classical result by Caruso (for $C_p$) to the non-cyclic case, demonstrating that the structure of the coefficient ring and the module structure of projective spaces impose strict constraints on equivariant operations.

4. Significance

• Structural Insight: The paper provides the first complete multiplicative description of the RO($C_p \times C_p$)-graded coefficient ring for both odd and even primes. This is a foundational step for equivariant stable homotopy theory, as the coefficient ring serves as the base for all other computations.
• Power Operations: The results are directly applicable to the study of power operations on equivariant $E_\infty$-ring spectra (like $MU_G$ and $H\mathbb{F}_p$). The cohomology of classifying spaces of symmetric groups governs these operations, and the computed rings provide the necessary algebraic data.
• Methodological Advancement: The successful application of the Tate square to $C_p \times C_p$ demonstrates a robust framework for handling non-cyclic groups, moving beyond the reliance on tools specific to cyclic groups (like the specific properties of the Tate square for $C_p$).
• Obstruction Theory: The non-liftability result highlights the subtle differences between cyclic and non-cyclic groups in equivariant cohomology operations, showing that the "lifting" problem is more restrictive in the presence of multiple non-trivial subgroups.

In summary, this work fills a critical gap in equivariant homotopy theory by solving the coefficient ring problem for the smallest non-cyclic abelian group, providing explicit algebraic structures that will serve as a reference for future computations in the field.