# Doubts on obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $$4$$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $$X$$ admits a Postnikov tower of principal fibrations, which is a Postnikov tower

$$\require{AMScd}$$ $$\begin{CD} \vdots & & \vdots\\ @| & @VVV \\ X @>>> X_3 \\ @| & @VVV \\ X @>>> X_2\\ @| & @VVV\\ X @>>> X_1 \end{CD}$$ where any map $$X_n \rightarrow X_{n-1}$$ is such that exists a fibration sequence $$F\rightarrow E\rightarrow B$$ and weak homotopy equivalences $$X_n\rightarrow F$$ and $$X_{n-1}\rightarrow E$$ such that

$$\require{AMScd}$$ $$\begin{CD} X_n @>>> X_{n-1} \\ @VVV & @VVV \\ F @>>> E@>>> B\\ \end{CD}$$ commutes.
The problem is that when a few pages later he deals with the problem of finding a map $$W\rightarrow X_n$$ to complete the commutative diagram $$\require{AMScd}$$ $$\begin{CD} A @>>> X_n \\ @VVV & @VVV \\ W @>>> X_{n-1}\\ \end{CD}$$ where $$(W,A)$$ is a $$CW$$ pair and $$A\rightarrow W$$ is the inclusion, he supposes that $$X_n$$ actually is the pullback of the path fibration $$PB\rightarrow B$$ via $$X_{n-1}\rightarrow B$$. Why can he suppose this?

• The pullback you've described is the definition of the homotopy fiber of the map $X_{n-1}\to B$. One may show in general that if $F\to E\to B$ is a fibration sequence, then $F$ is homotopy equivalent to the homotopy fiber of $E\to B$. Now apply this to the fibration sequence $X_n\to X_{n-1}\to B$. – Christian Carrick Oct 24 '18 at 5:26