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.
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