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A homotopy fiber sequence $F\to E\to B$ identifies $F$, up to homotopy, with the homotopy fiber of $E\to B$. It induces a long exact sequence of homotopy groups.

Let $A[n]$ be the nonnegative chain complex having $A$ in degree $n$, zero in every other degree, and zero differential. The simplicial Eilenberg–MacLane space isIts underlying simplicial set is a Kan complex, with ${\pi }_{n}K(A,n)\cong A$ and all other positive homotopy groups zero.

Solved by gpt-5.6-sol high.

Let $P$ be the homotopy fiber of the map representingThe long exact sequence of homotopy groups shows that every ${\pi }_i(P)$ except ${\pi }_1(P)$ vanishes and thatHence $P$ is either $K({\mathbb{F}}_2\oplus {\mathbb{F}}_2,1)$ or $K(\mathbb{Z}/4,1)$.

The extension is classified by the degree-two class represented by the original map. If $t\in H^1(K({\mathbb{F}}_2,1);{\mathbb{F}}_2)$ is the standard generator, that class isin $H^{\ast }({\mathbb{RP}}^{\infty };{\mathbb{F}}_2)={\mathbb{F}}_2[t]$. It therefore classifies the non-split extension, whose middle group is $\mathbb{Z}/4$. Thus

Solved by gpt-5.6-sol high.