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I'm trying to understand the usefulness of "substituting" a continuous map f , by a fibration F. By substituting, I mean there is the result that given a continuous map $f:X \rightarrow Y $ , for any topological spaces $X,Y$, one can define a new space $X'$, and a fibration $F : X' \rightarrow Y$ , so that that the maps $f, F$ (together with the identity map $id: Y \rightarrow Y$) satisfy the commutative square $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} X' & \ra{F} & Y \\ \da{d} & & \da{=} \\ X & \ras{f} & Y \\ \end{array} $$ I know that once we have a fibration $F: X \rightarrow Y $ , we have two new conditions satisfied by $F$ that are not satisfied by $f$:

i) The fibers of $F$ are homotopically-equivalent to each other, and,

ii) A fibration $F$ gives rise to a long-exact sequence of homotopy groups.

So we may gain some information about homotopy groups from ii). Still: what can I really learn about $f: X \rightarrow Y $ from this associated fibration ? How is this associated fibration useful in general?

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  • $\begingroup$ Well, a minor point is that it is part of one of the necessary axioms in order for $\mathbf{Top}$ to be a model category en.wikipedia.org/wiki/Model_category. $\endgroup$
    – Dan Rust
    Jan 30 '14 at 0:39
  • $\begingroup$ I hope you don't mind, I added in the commutative square you mentioned (and called the canonical deformation retraction $d$). $\endgroup$
    – Dan Rust
    Jan 30 '14 at 0:47
  • $\begingroup$ No problem, Daniel Rust; thanks, actually. $\endgroup$ Jan 30 '14 at 0:57
  • $\begingroup$ @HansLundmark: Thanks ;). $\endgroup$ Feb 27 '14 at 0:46
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Some thoughts:

1) The Serre spectral sequence is surely one of the most useful tools one has in topology - many topological problems can be tackled by first hoping that something is a fibration and applying the Serre spectral sequence.

2) The compulsion to replace a map by a fibration (or cofibration) is explained well on: http://math.uoregon.edu/~ddugger/hocolim.pdf page 6 where Dugger explains the more general situation of homotopy colimits. This is related because, basically, one way to compute homotopy limits is to replace maps by fibrations (actually he discusses homotopy colimits but the theory is dual) first in order to get a limit that is homotopy invariant and ``tractable" via the methods of homotopy theory.

3) Point ii) really buys us quite a lot: say I want to show that two CW complexes $E$ and $E'$ are homotopy equivalent. Then I am reduced to showing that they have isomorphic homotopy groups by Whitehead's theorem, and then I can pray that there is a comparison map between the two spaces and that they each sit in fiber sequences --- standard homological algebra techniques then apply.

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  • $\begingroup$ In 3), "Then I am reduced to showing that they have isomorphic homotopy groups by Whitehead's theorem" is false, as stated. (Your next line indicates that you have the correct version in mind, but it might be worth rephrasing for the benefit of others.) $\endgroup$
    – user64687
    Jan 30 '14 at 9:24

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