3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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 $X_n\to X_{n-1}\to B$. $\endgroup$ – Christian Carrick Oct 24 '18 at 5:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.