tmf Is Not a Ring Spectrum Quotient of String Bordism

Abstract

This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}⟨8⟩[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of ${\pi }_{\ast }\mathrm{MO}⟨8⟩$, the ${\pi }_{\ast }\mathrm{MO}⟨8⟩$-module $${\pi }_{\ast }(\mathrm{MO}⟨8{⟩}_{(p)}/X)$$ is not (even abstractly) isomorphic to ${\pi }_{\ast }{\mathrm{tmf}}_{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of ${\pi }_{\ast }(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $${\pi }_{\ast }(R/X)\otimes \mathbf{Q}\cong {\mathrm{H}}_{\ast }(\mathrm{Tot}(\mathrm{K}(X)))\otimes \mathbf{Q},$$ where the right-hand side is the rational homology of the (total) Koszul complex of $X$, which is strictly bigger than ${\pi }_{\ast }(R)/(X)\otimes \mathbf{Q}$ unless $X$ is a ${\pi }_{\ast }(R)\otimes \mathbf{Q}$-quasi-regular sequence. The result then follows from the fact that the kernel of the $p$-local Witten genus cannot be generated by a ${\pi }_{\ast }\mathrm{MO}⟨8⟩\otimes \mathbf{Q}$-quasi-regular sequence.

9 pages; statement of the new Theorem 3 made clearer

References in corpus (2)

• Algebraic topology and modular forms
• The Cayley plane and String bordism