We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

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

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

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

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

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