[Paper Review] Quasi-elliptic cohomology and its Spectrum

Zhen Huan|arXiv (Cornell University)|Mar 20, 2017

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper constructs an orthogonal G-spectrum representing quasi-elliptic cohomology by introducing a new category of orthogonal G-spectra, GwS, using homotopical right adjoints to fixed point functors. The construction applies broadly to equivariant cohomology theories like Tate K-theory and generalized Morava E-theories, offering a new framework for global homotopy theory that may underlie global elliptic cohomology.

ABSTRACT

Ginzburg, Kapranov and Vasserot conjectured the existence of equivariant elliptic cohomology theories. In this paper, to give a description of equivariant spectra of the theories, we study an intermediate theory, quasi-elliptic cohomology. We formulate a new category of orthogonal G-spectra and construct explicitly an orthogonal G-spectrum of quasi-elliptic cohomology in it. The idea of the construction can be applied to a family of equivariant cohomology theories, including Tate K-theory and generalized Morava E-theories. Moreover, this construction provides a functor from the category of global spectra to the category of orthogonal G-spectra. In addition, from it we obtain some new idea what global homotopy theory is right for constructing global elliptic cohomology theory.

Motivation & Objective

• To address the challenge of constructing orthogonal G-spectra for equivariant elliptic cohomology theories, which are too complex for direct construction.
• To develop a new category of orthogonal G-spectra, GwS, that supports the construction of spectra for intermediate theories like quasi-elliptic cohomology.
• To explore whether the category GwS is suitable for housing equivariant elliptic spectra and for building global elliptic cohomology theories.
• To establish a functor from global spectra to orthogonal G-spectra, advancing the understanding of global homotopy theory.
• To provide a concrete model for ultra-commutative global cohomology theories via quasi-elliptic cohomology, inspired by Ganter's suggestion.

Proposed method

• Introduces a new category of orthogonal G-spectra, GwS, to handle the lack of right adjoints to fixed point functors.
• Uses homotopical right adjoints of fixed point functors to construct the representing orthogonal G-spectrum for quasi-ellictic cohomology.
• Applies the construction to theories of the form $ QE^{*}_{G}(X) = \left(\prod_{\sigma \in G^{\text{tors}}} E^{*}_{\Lambda(\sigma)}(X^{\sigma})\right)^{G} $, where $ E $ has K-theory-like features.
• Defines structure maps $ \mu^{QE}_{(g,h)} $ via convex combinations involving norms of orthogonal components, ensuring associativity and centrality.
• Establishes the spectrum structure via explicit formulas involving $ v_1, v_2 $ and $ w_1, w_2 $, with basepoint conditions when $ \|v_2\|^2 + \|w_2\|^2 \geq 1 $.
• Verifies monoidal axioms (associativity, unit, symmetry) using explicit path-lifting and norm-based interpolation in the construction.

Experimental results

Research questions

• RQ1 Can an orthogonal G-spectrum be constructed for quasi-elliptic cohomology despite the absence of right adjoints to fixed point functors?
• RQ2 Is the category GwS a suitable framework for representing equivariant cohomology theories like Tate K-theory and generalized Morava E-theories?
• RQ3 Does the construction of quasi-elliptic cohomology spectra via homotopical adjunctions provide a viable path toward global elliptic cohomology?
• RQ4 Can the resulting orthogonal G-spectrum be realized as an underlying orthogonal spectrum, or is a new global homotopy theory required?
• RQ5 How does the monoidal structure of the quasi-elliptic cohomology spectrum behave under composition and symmetry?

Key findings

• An orthogonal G-spectrum for quasi-elliptic cohomology is explicitly constructed in the category GwS using homotopical right adjoints.
• The construction generalizes to any equivariant cohomology theory $ E $ with K-theory-like properties, yielding spectra for $ QE^{*}_{G}(X) = \left(\prod_{\sigma} E^{*}_{\Lambda(\sigma)}(X^{\sigma})\right)^{G} $.
• The spectrum structure satisfies all monoidal axioms: associativity, unit centrality, and symmetry, verified via norm-based interpolation formulas.
• The resulting orthogonal G-spectrum is not the underlying orthogonal spectrum of any orthogonal spectrum, indicating a need for a new global homotopy theory.
• The construction provides a functor from global spectra to orthogonal G-spectra, suggesting a pathway toward ultra-commutative global cohomology theories.
• The monoidal structure is defined via convex combinations: $ (1 - \sqrt{\|v_2\|^2 + \|w_2\|^2}) \cdot \mu_F + \sqrt{\|v_2\|^2 + \|w_2\|^2} \cdot (v_2 + w_2) $, with basepoint when the norm exceeds 1.