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I'm reading May's More Concise Algebraic Topology and the first half of the book seems to be written under the assumption that the reader has the motivation that we want to localize the underlying topological space when doing computations. Coming from reading his prior book, this seems like a lot more highly technical machinery with no goal in sight. Where does the motivation for topological localization come from?

Naiche Cimarron Downey
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    I am not qualified to say more about your specific line of question, but in algebraic number theory and algebraic geometry and many other subjects, questions which are too hard to answer "globally" may often admit "local" answers... where some information has been thrown away (forgetful functors, if you like). And, naturally, there is the question of whether the aggregate of all (?) local answers determines the global one. (In number theory a keyword here is "Hasse Principle"... which holds sometimes, but fails sometimes, and measuring failure is interesting...) – paul garrett Jul 23 '18 at 22:20
  • It might be helpful to read the preface of Sullivan's book (since I guess he originally came up with the whole shabang) . I don't really have sufficient background to fully understand the discussion. – Andres Mejia Jul 23 '18 at 22:41
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    in these notes Adams claims that much of the motivation came from work of Serre. – Andres Mejia Jul 23 '18 at 22:53
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    It seems that the quote “Il y a l`a la possibilit´e d’une ´etude locale (au sens arithm´etique!) des groupes d’homotopie . . . ” J-P. Serre,

    makes the Hasse Principle and related stuff the motivation. I actually saw this in practice in an REU in chromatic homotopy theory a year ago but didn't make the connection between the computations I was doing there and topological localisations via postnikov towers.

     – Naiche Cimarron Downey Jul 23 '18 at 23:46
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    The motivation for localisation in my work is normally as simple as being able to provide a partial answer to a problem too complex to solve with current techniques. Although the integral problem may be unapproachable it is often the case that something can be said after localisation at one or more primes. Indeed, we often end up with ridiculuous statements that completely determine the localised homotopy type of certain mapping spaces at all primes, whilst saying absolutely nothing about the integral homotopy type. – Tyrone Jul 24 '18 at 11:14
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    @Tyrone Oh wow I thought that usually knowing what happened at all primes was considered almost a complete understanding. Are you saying that we really can't often deduce $X \simeq Y$ when it is true at every prime? – Connor Malin Mar 16 '21 at 19:08
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    @ConnorMalin we cannot. The projection $\pi:Sp_2\rightarrow S^7$ is locally-trivial with fibre $Sp_1\cong S^3$. Let $E_5$ be the pullback of $\pi$ and the degree $5$ map on $S^7$ (this space is theHillton-Roitberg criminal). Compare $E_5$ and $Sp_2$. (The Mislin Genus $G(X)$ of a nilpotent complex $X$ is the set of all homotopy types $Y$ with $Y_{(p)}\simeq X_{(p)}$ for each prime $p$. This was introduced by Mislin and studied by Hilton, McGibbon, etc. The problem is no more/less difficult than the corresponding algebraic problem. I'll be happy to dig out some references if you like.) – Tyrone Mar 16 '21 at 19:48

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