# Topology Seminar

## tmf Is Not a Ring Spectrum Quotient

Carl McTague, Rochester

Wednesday, September 6th, 2017
3:40 PM - 4:40 PM
Hylan 1106A

To begin, I will describe a new result on the greatest common divisor of binomial coefficients ${n \choose q}, {n \choose 2q}, {n \choose 3q},\dots$ A result related to the geometry of manifolds, answering a question raised the last time I spoke in this seminar, and recently published in the Amer. Math. Monthly.

Next, I will describe a new proof that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $\pi_*\mathrm{MO}\langle8\rangle$, the $\pi_*\mathrm{MO}\langle8\rangle$-module

is not (even abstractly) isomorphic to $\pi_*\mathrm{tmf}_{(p)}$. The key is showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $\pi_*(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces

where the right-hand side is the rational homology of the (total) Koszul complex of $X$, which is strictly bigger than $\pi_*(R)/(X)\otimes\mathbf{Q}$ unless $X$ is a $\pi_*(R)\otimes\mathbf{Q}$-quasi-regular sequence. The result then follows from the fact that the kernel of the $p$-local Witten genus cannot be generated by a $\pi_*\mathrm{MO}\langle8\rangle\otimes\mathbf{Q}$-quasi-regular sequence.

Event contact: carl dot mctague at rochester dot edu