Topological Mathieu Moonshine

Understand

We explore the Atiyah-Hirzebruch spectral sequence for the $tmf^\bullet[\frac12]$-cohomology of the classifying space $BM_{24}$ of the largest Mathieu group $M_{24}$, twisted by a class $\omega \in H^4(BM_{24};Z[\frac12]) \cong Z_3$.

• Our exploration includes detailed computations of the $Z_3$-cohomology of $M_{24}$ and of the first few differentials in the AHSS.
• We are specifically interested in the value of $tmf^\bullet_\omega(BM_{24})[\frac12]$ in cohomological degree $-27$.
• Our main computational result is that $tmf^{-27}_\omega(BM_{24})[\frac12] = 0$ when $\omega \neq 0$.