We know that there is a fiber sequence: $$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$

Is this fiber sequence induced from a short exact sequence?

• If so, is that $$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

• Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

• So $\mathbb{CP}^{\infty}$ is a normal subgroup of $\mathrm{String}$, so $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $\mathrm{Spin}$ is a quotient group of $\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!