  Recently it has been pointed out that the skyrmions carry two independent
topology, the baryon topology and the monopole topology. We provide more
evidence to support this. In specific, we prove that the baryon number $B$ can
be decomposed to the monopole number $m$ and the shell number $n$, so that $B$
is given by $B=mn$. This tells that the skyrmions may more conveniently be
classified by two integers $(m,n)$. This is because the rational map which
determines the baryon number in the popular multi-skyrmion solutions actually
describes the monopole topology $\pi_2(S^2)$ which is different from the baryon
topology $\pi_3(S^3)$. Moreover, we show that the baby skyrmions can also be
generalized to have two topology, $\pi_1(S^1)$ and $\pi_2(S^2)$, and thus
should be classified by two topological numbers $(m,n)$. Furthermore, we show
that the vacuum of the Skyrme theory can be classified by two topological
numbers $(p,q)$, the $\pi_1(S^1)$ of the sigma-field and the $\pi_3(S^2)$ of
the normalized pion-field. This means that the Skyrme theory has multiple vacua
similar to the Sine-Gordon theory and QCD combined together. This puts the
Skyrme theory in a totally new perspective. We discussthe physical implications
of our results.
