[Paper Review] Actions of Eilenberg-MacLane spaces on K-theory spectra and uniqueness of twisted K-theory

Benjamin Antieau, David Gepner+1 more|arXiv (Cornell University)|Jun 25, 2011

Homotopy and Cohomology in Algebraic Topology15 references5 citations

TL;DR

This paper establishes the uniqueness of twisted K-theory in both real and complex cases by proving that all maps from Eilenberg-MacLane spaces $K(\mathbb{Z},3)$ to $BGL_1K$ and $K(\mathbb{Z}/2,2)$ to $BGL_1KO$ are determined up to homotopy by endomorphisms of the source spaces, using computations of $K$-theory of Eilenberg-MacLane spaces. The key result is that the space of such maps is isomorphic to $\mathbb{Z}$ and $\mathbb{Z}/2$ respectively, ensuring all reasonable definitions of twisted K-theory agree.

ABSTRACT

We prove the uniqueness of twisted K-theory in both the real and complex cases using the computation of the K-theories of Eilenberg-MacLane spaces due to Anderson and Hodgkin. As an application of our method, we give some vanishing results for actions of Eilenberg-MacLane spaces on K-theory spectra.

Motivation & Objective

• To establish the uniqueness of twisted K-theory across different definitions in the complex and real cases.
• To classify maps from Eilenberg-MacLane spaces $K(\mathbb{Z},3)$ and $K(\mathbb{Z}/2,2)$ to the classifying spaces $BGL_1K$ and $BGL_1KO$ of units of $K$-theory spectra.
• To show that all such maps are determined up to homotopy by endomorphisms of the source spaces, ensuring consistency across definitions.
• To provide vanishing results for actions of Eilenberg-MacLane spaces on $K$-theory spectra using spectral computations.

Proposed method

• Uses the splitting of $GL_1K$ into $K(\mathbb{Z}/2,0) \times K(\mathbb{Z},2) \times BSU_\otimes$ to analyze maps from $K(\mathbb{Z},3)$ to $BGL_1K$.
• Applies the computation of $K$-theory of Eilenberg-MacLane spaces by Anderson and Hodgkin to determine the group of maps from $K(\mathbb{Z},3)$ to $BGL_1K$.
• Shows that the component $BBSU_\otimes$ contributes trivially to $[K(\mathbb{Z},3), BBSU_\otimes]$, implying maps factor through $K(\mathbb{Z},3)$.
• Uses the delooping of the infinite loop space splitting to relate $BGL_1K$ to $K(\mathbb{Z}/2,1) \times K(\mathbb{Z},3) \times BBSU_\otimes$.
• Applies the action of $PU(\mathcal{H}) \simeq K(\mathbb{Z},3)$ on Fredholm operators to construct a spectrum-level action on Joachim's $K$-theory spectrum.
• Demonstrates that the conjugation action of $PU(\mathcal{H})$ on Joachim's spectrum induces an $A_\infty$-map to $GL_1K$, delooping to a map $K(\mathbb{Z},3) \to BGL_1K$.

Experimental results

Research questions

• RQ1 Are all maps from $K(\mathbb{Z},3)$ to $BGL_1K$ homotopic to maps that factor through $K(\mathbb{Z},3)$?
• RQ2 What is the group of homotopy classes of maps $[K(\mathbb{Z},3), BGL_1K]$?
• RQ3 How do different definitions of twisted $K$-theory relate when they arise from maps into $BGL_1K$?
• RQ4 What is the role of the $BSU_\otimes$ component in classifying maps from $K(\mathbb{Z},3)$ to $BGL_1K$?
• RQ5 Can the action of $PU(\mathcal{H})$ on Fredholm operators be used to construct a well-defined twisted $K$-theory spectrum?

Key findings

• The group $[K(\mathbb{Z},3), BGL_1K]$ is isomorphic to $\mathbb{Z}$, with the isomorphism induced by the endomorphism ring of $K(\mathbb{Z},3)$.
• The group $[K(\mathbb{Z}/2,2), BGL_1KO]$ is isomorphic to $\mathbb{Z}/2$, showing uniqueness up to automorphism in the real case.
• The space $[K(\mathbb{Z},3), BBSU_\otimes]$ vanishes, implying that maps from $K(\mathbb{Z},3)$ to $BGL_1K$ factor through $K(\mathbb{Z},3)$ up to homotopy.
• The conjugation action of $PU(\mathcal{H})$ on Joachim's $K$-theory spectrum induces a well-defined $A_\infty$-map $K(\mathbb{Z},3) \to BGL_1K$, realizing the standard twist.
• All reasonable definitions of twisted $K$-theory arise from maps that are homotopic to a fixed standard map, ensuring uniqueness.
• Vanishing results for actions of Eilenberg-MacLane spaces on $K$-theory spectra are obtained via spectral computations, particularly in the $BSU_\otimes$ component.