[Paper Review] The $RO(C_4)$ integral homology of a point

Nick Georgakopoulos|arXiv (Cornell University)|Dec 14, 2019

Homotopy and Cohomology in Algebraic Topology4 references4 citations

TL;DR

This paper computes the RO(C₄)-graded integral homology of a point as a Green functor, using a combination of theoretical algebraic topology and a custom computer program. It shows that the homology is generated—up to group extensions—by Euler and orientation classes of the irreducible real C₄-representations σ and λ, with the exception of a Z/4 summand in degree −3(S⁻²λ), which arises from a non-split extension of Z/2 groups that are themselves generated by these classes.

ABSTRACT

We compute the $RO(C_4)$ integral homology of a point with complete information as a Green functor, and we show that it is generated, in a slightly generalized sense, by the Euler and orientation classes of the irreducible real $C_4$-representations. We have devised a computer program that automates these computations for groups $G=C_{p^n}$ and we have used it to verify our results for $G=C_4$ in a finite range.

Motivation & Objective

• To compute the full RO(C₄)-graded integral homology of a point as a Green functor.
• To determine the algebraic structure of π^{C₄}_⋆(HZ) in terms of generators and relations.
• To develop and validate a computational tool for RO(G)-graded homology computations for G = Cₚⁿ.
• To resolve subtleties in equivariant homotopy theory, particularly regarding torsion and extension classes in Mackey functors.
• To compare and reconcile results with prior work, especially Zeng (2017), using a generator-based multiplicative description.

Proposed method

• Uses equivariant cellular chains for positive virtual representations and Spanier-Whitehead duality for negative ones.
• Applies three spectral sequences—two Atiyah-Hirzebruch and one Kunneth—on Mackey functor-valued chain complexes.
• Employs a computer program to automate additive and multiplicative structure computations for G = Cₚⁿ with bounded dimension.
• Performs calculations in the symmetric monoidal category of Z-modules to handle Tor terms in the Kunneth spectral sequence.
• Uses denominator-clearing and Frobenius relations to simplify products and verify relations.
• Relies on the Gold relation and known exact sequences to deduce complex multiplicative relations.

Experimental results

Research questions

• RQ1 What is the complete RO(C₄)-graded integral homology of a point as a Green functor?
• RQ2 How can the multiplicative structure of π^{C₄}_⋆(HZ) be described using only Euler and orientation classes?
• RQ3 What is the role of the Z/4 summand in H^{C₄}_{-3}(S^{-2λ}; Z), and how does it relate to the standard generators?
• RQ4 To what extent can such computations be automated for Cₚⁿ with n > 2?
• RQ5 How do the results compare with Zeng’s (2017) Tate-square-based computation for RO(Cₚ²)-graded homology?

Key findings

• The RO(C₄)-graded integral homology of a point is generated, in a generalized sense, by the Euler class aσ and orientation class aλ of the irreducible real C₄-representations.
• The homology group HC₄^{−3}(S^{-2λ}; Z) ≅ Z/4 is not generated by aσ and aλ alone, but fits into a non-split short exact sequence 0 → Z/2 → Z/4 → Z/2 → 0.
• The Z/2 summands in this extension are realizable using only aσ and aλ, so the full homology is generated when group extensions are included as operations.
• The computer program successfully verifies the results in a finite range and caught errors in early drafts, demonstrating its utility as a verification tool.
• The multiplicative structure is fully described by relations among aσ, aλ, and their quotients, with key relations derived from the Gold relation and Frobenius identities.
• The computation reveals that products involving uλ and certain quotients vanish due to torsion constraints, such as uλ · (2aλ/aσ) = 0.