In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{ r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic piece containing the bottom cell $T^{ 2n+1}\{ p^r\}$ is null homotopic for $n>1$, $p>3$ and $r>1$. This result strengthens the classical result that $\Omega T^{ 2n+1}\{ p^r\}$ has an exponent $p^{ r+1}$.
Categories: math.AT math.GR