Topological $ΔG$ homology of rings with twisted $G$-action

This paper constructs a unified framework of topological $\Delta G$-homology for rings with twisted $G$-action, which generalizes existing theories like THH and THR, introduces new examples such as quaternionic THH, and establishes homotopical identifications with twisted free loop spaces via a novel family of crossed simplicial groups.

The Gist

Imagine you are a master architect trying to build a universal measuring tape for shapes and structures. In the world of mathematics, specifically in a field called algebraic topology, mathematicians have been trying to measure the "shape" of algebraic objects (like rings) for decades.

This paper, written by Angelini-Knoll, Merling, and Pérour, introduces a new, super-flexible measuring tape called Topological $\Delta G$-Homology.

Here is the breakdown of what they did, using everyday analogies:

1. The Problem: One Size Doesn't Fit All

Imagine you have a box of different toys:

• Some toys are just rings (like a standard hula hoop).
• Some are rings with a mirror (if you flip them, they look the same, but backwards).
• Some are rings that spin (they have a built-in motor).

For a long time, mathematicians had specific rulers for specific toys:

• THH (Topological Hochschild Homology): A ruler for standard rings. It measures how the ring twists around a circle ($S^1$).
• THR (Real THH): A ruler for rings with mirrors. It measures how they behave when flipped and rotated ($O(2)$).

But what if you have a ring that spins in a weird, 4-step pattern (like a quaternion)? Or a ring that has a mix of spinning and flipping? The old rulers didn't work. You needed a universal ruler that could handle any combination of these symmetries.

2. The Solution: The "Twisted G-Action"

The authors realized that all these different behaviors (spinning, flipping, doing nothing) can be described by a single concept: A Ring with a Twisted G-Action.

Think of a group $G$ as a set of instructions (like "Spin," "Flip," or "Do Nothing").

• Even instructions: If the instruction is "Spin," the ring behaves normally.
• Odd instructions: If the instruction is "Flip," the ring behaves like a mirror image (it reverses the order of multiplication).

The authors created a framework where a ring can follow any set of these instructions, whether they come from a simple circle, a mirror, or a complex 4-step dance.

3. The Tool: "Crossed Simplicial Groups" (The Lego Bricks)

To build their new ruler, they needed a new type of Lego brick.

• Standard math uses Simplicial Sets (like triangles and tetrahedrons) to build shapes.
• The authors invented Crossed Simplicial Groups.

The Analogy: Imagine you are building a tower with Lego bricks.

• In the old way, you could only stack bricks straight up.
• In this new way, the bricks have magic hinges. When you stack them, the bricks can rotate, flip, or swap places with each other based on a secret code (the group $G$).
• This allows them to build complex, twisting structures that represent the "shape" of these weird rings.

4. The New Inventions: THQ and Others

Using this new framework, they didn't just generalize the old rulers; they invented brand new ones:

• THQ (Quaternionic Topological Hochschild Homology): This is the star of the show. It measures rings that follow a 4-step dance (related to quaternions, a type of number system used in 3D graphics).

  • The Metaphor: If THH is a hula hoop spinning, and THR is a hula hoop being flipped, THQ is a hula hoop that is spinning, flipping, and then spinning in a completely different dimension. It has a "Pin(2)" symmetry, which is like a super-complex dance move.

• Topological Twisted Symmetric Homology: A ruler for rings that can swap their parts in any order, but with a twist (some swaps flip the ring, others don't).

5. The Big Discovery: The "Loop Space" Connection

The most exciting part of the paper is what happens when they apply these rulers to Loop Spaces (imagine a rubber band stretched around a shape).

They proved a beautiful connection:

• If you take a shape, put a rubber band around it, and measure it with their new Quaternionic Ruler (THQ), the result is exactly the same as taking a Twisted Free Loop Space.

The Analogy:
Imagine you have a piece of string (the loop).

• Standard Loop: You tie the ends together.
• Twisted Loop: You twist the string 180 degrees before tying the ends.
• The Theorem: The authors proved that measuring the "Quaternionic Ring" of a shape is mathematically identical to measuring the "Twisted String" of that shape.

This is huge because it translates a very abstract algebraic problem (measuring rings) into a geometric one (measuring twisted strings), which is often much easier to visualize and calculate.

Summary

• What they did: They built a universal mathematical framework to measure rings that have complex, mixed symmetries (spinning, flipping, etc.).
• How they did it: By inventing a new type of "Lego brick" (Crossed Simplicial Groups) that can twist and flip.
• Why it matters:
  1. It unifies many different existing theories into one big family.
  2. It creates new tools (like THQ) to study 4-dimensional symmetries.
  3. It connects abstract algebra to geometry, showing that measuring these rings is the same as measuring twisted loops of string.

In short, they took a messy pile of different measuring tools, realized they were all variations of the same thing, built a master tool to handle them all, and discovered that this master tool reveals a hidden link between algebra and twisted strings.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the need for a unified framework to generalize Topological Hochschild Homology (THH) and its variants (such as Real THH and Cyclic Homology) to rings equipped with more complex symmetries than standard group actions or anti-involutions.

• Context: Classical Hochschild homology and Cyclic homology are fundamental in algebraic K-theory. Their topological counterparts (THH, Topological Cyclic Homology) are crucial for computing algebraic K-theory of ring spectra via trace methods.
• Limitation of Existing Theories:
  • Standard THH corresponds to rings with trivial action (or $S^1$-action on the Hochschild complex).
  • Real THH (THR) corresponds to rings with anti-involution (action by $C_2$ where the non-trivial element acts as an anti-homomorphism).
  • Existing frameworks struggle to simultaneously handle rings where different group elements act as homomorphisms and others as anti-homomorphisms in a coherent way, particularly for groups like $C_4$ or general crossed simplicial groups.
• Goal: Construct a general homotopical invariant, Topological $\Delta G$-Homology, for "rings with twisted $G$-action," which generalizes both rings with $G$-action and rings with anti-involution. This includes new invariants like Quaternionic THH (THQ) and Topological Twisted Symmetric Homology.

2. Methodology and Technical Framework

The authors employ a combination of combinatorial category theory, operad theory, and $\infty$-category theory to build their construction.

A. Crossed Simplicial Groups and Parity

• Crossed Simplicial Groups ($\Delta G$): The authors utilize the theory of crossed simplicial groups (introduced by Fiedorowicz and Loday), which extend the simplex category $\Delta$ by adding automorphism groups $G_n$ at each level $n$.
• Groups with Parity: A key innovation is the introduction of a group with parity $(G, \phi)$, where $\phi: G \to \{1, -1\}$ is a homomorphism. Elements mapping to $1$ are "even" (acting as homomorphisms), and elements mapping to $-1$ are "odd" (acting as anti-homomorphisms).
• New Construction ($\Delta_\phi \wr \Sigma$): They construct a new family of crossed simplicial groups, $\Delta_\phi \wr \Sigma$, associated with any group with parity. This generalizes the hyperoctahedral crossed simplicial group.
  • If $\phi$ is trivial, it recovers the symmetric crossed simplicial group ($\Delta \Sigma$).
  • If $\phi = \text{id}_{C_2}$, it recovers the hyperoctahedral group ($\Delta H$).

B. Twisted $G$-Actions and Operads

• Twisted $G$-Rings: A ring spectrum with twisted $G$-action is defined as an associative ring spectrum where even elements of $G$ act by ring homomorphisms and odd elements act by ring anti-homomorphisms.
• Operadic Characterization: The authors prove that rings with twisted $G$-action are precisely algebras over a specific twisted operad, denoted $\text{Assoc}_\phi$. This operad is a semi-direct product of the associative operad and the group $G$, encoding the parity constraints.
• Symmetric Monoidal Envelope: They establish an isomorphism of symmetric monoidal categories between the symmetric monoidal envelope of $\text{Assoc}_\phi$ and the category $\Delta_\phi \wr \Sigma_+$. This allows them to view twisted $G$-rings as symmetric monoidal functors from $\Delta_\phi \wr \Sigma_+$ to the category of spectra.

C. The Homology Construction

• Bar Construction: They define a $\Delta G$-bar construction for a ring spectrum $R$ with twisted $G$-action. This is a functor from the crossed simplicial group to spectra.
• Self-Dual Case: For self-dual crossed simplicial groups (where there is an equivalence $\Delta G^{op} \simeq \Delta G$), they define:
  • Topological $\Delta G$-Homology ($THG(R)$): The geometric realization of the bar construction.
  • Variants: They define topological negative cyclic ($TG^-$), periodic cyclic ($TG^{per}$), and positive cyclic ($TG^+$) homology as homotopy orbits, fixed points, and Tate constructions of $THG(R)$ with respect to the geometric realization of the group $|G_q|$.
• General Case: For non-self-dual groups, they define Positive Topological $\Delta G$-Homology ($TG^+$) directly as the colimit of the bar construction, interpreting it as a form of homotopical functor homology.

3. Key Contributions and Results

A. New Invariants

1. Quaternionic Topological Hochschild Homology (THQ):

   • Associated with the quaternionic crossed simplicial group $\Delta Q$.
   • Input: Ring spectra with twisted $C_4$-action (where the generator $t$ satisfies $t^4 \simeq \text{id}$, with $t$ and $t^3$ acting as anti-homomorphisms).
   • Structure: $THQ(R)$ carries a $\text{Pin}(2)$-action.
   • Significance: This is a new invariant distinct from Real THH, conjectured to be the norm from $C_4$ to $\text{Pin}(2)$.

2. Topological Twisted Symmetric Homology ($T_\phi$):

   • Associated with $\Delta_\phi \wr \Sigma$.
   • Generalizes topological symmetric homology and hyperoctahedral homology.
   • Provides a homotopical analogue of algebraic results by Fiedorowicz, Ault, and Graves.

B. Main Theorems

• Theorem A (Construction): For a self-dual crossed simplicial group $\Delta G$, there exists a topological homology $THG(R)$ carrying a $|G_q|$-action for any ring spectrum $R$ with twisted $G_0$-action.

  • Recovers $THH$ (for $\Delta C$), $THR$ (for $\Delta D$), and yields $THQ$ (for $\Delta Q$).

• Theorem B (Loop Space Identification for THQ):
  Let $X$ be a space with $C_4$-action. There is an equivalence of spectra with left $\text{Pin}(2)$-action:
  $$THQ(\Sigma^\infty_+ \Omega_q X) \simeq \Sigma^\infty_+ L_\tau X$$
  where $L_\tau X = \{ \gamma: [0,1] \to X \mid t^2 \gamma(0) = \gamma(1) \}$ is the twisted free loop space. This identifies $THQ$ of a loop space with a twisted free loop space, analogous to how $THH$ relates to the free loop space.

• Theorem C (General Positive Homology): For any crossed simplicial group $\Delta G$, there is a positive topological homology $TG^+(R)$. If $\Delta G$ is self-dual, $TG^+(R)$ is equivalent to the $|G_q|$-homotopy orbits of $THG(R)$.

• Theorem D (Twisted Symmetric Homology of Loop Spaces):
  Let $\phi: G \to C_2$ be a parity. For a connected space $X$ with $G$-action:
  $$T_\phi(\Sigma^\infty_+ \Omega_\phi X) \simeq (\Sigma^\infty_+ \Omega Q X)^{hG}$$
  This generalizes classical results of Fiedorowicz, Ault, and Graves from the derived category of modules to the stable homotopy category (spectra).

4. Significance and Impact

1. Unification: The paper provides a single combinatorial and categorical framework that unifies THH, THR, and various other homology theories under the umbrella of "twisted $G$-actions."
2. Norm Perspective: The authors interpret their constructions as combinatorial models for norms in equivariant homotopy theory.
   • $THH$ is the norm from the trivial group to $S^1$.
   • $THR$ is the norm from $C_2$ to $O(2)$.
   • $THQ$ is proposed as the norm from $C_4$ to $\text{Pin}(2)$.
     This perspective connects the algebraic construction to deep geometric concepts in equivariant stable homotopy theory.
3. New Computational Tools: By identifying $THQ$ of loop spaces with twisted free loop spaces, the paper opens new avenues for computing equivariant invariants using fixed-point theory and loop space geometry.
4. Future Directions:
   • Hyperreal Homology: The authors suggest refining $THQ$ to a genuine $\text{Pin}(2)$-spectrum, termed "hyperreal Hochschild homology," which could shed light on involutive Heegaard Floer homology.
   • Equivariant K-Theory: The framework is expected to aid in computing equivariant algebraic K-theory groups via trace methods.
   • Braid and Reflexive Homology: The methods extend to braid and reflexive crossed simplicial groups, suggesting new connections to representation homology and involutive homology.

In summary, this paper significantly advances the field of algebraic topology by constructing a robust, generalized homology theory for rings with twisted symmetries, establishing deep connections between operads, crossed simplicial groups, and equivariant stable homotopy theory, and providing concrete computations for new invariants like Quaternionic THH.