# Higher associativity of Moore spectra and $(p)$-local Adams conjecture.

# Higher associativity of Moore spectra and $(p)$-local Adams conjecture.

Not much is known about homotopy coherent ring structures of the Moore spectrum $M_p(i)$ (the cofiber of $p^i$ self-map on the sphere spectrum$S^0$), especially when $i > 1$. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called $A_n$ structures. The only known results are that $M_p(1)$ is $A_{p-1}$ and not $A_p$ and that $M_2(i)$ are at least $A_3$ for $i>1$. In this talk, techniques will be developed to get estimates of `higher associativity' structures on $M_p(i)$. In particular, it will be shown that, $M_p(i)$ admits $A_{p^i-1}$-structure for odd primes and $A_{2^{i-1}-1}$-structure when $p=2$. This result requires solving stable $p$-local Adams Conjecture. Work presented here is joint with N.Kitchloo.