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In Algebraic Topology, why do we want to localize spaces?
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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?
algebraic-topology
asked Jul 23 at 22:04
Naiche Cimarron Downey
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up vote
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down vote
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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?
algebraic-topology
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
2
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 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 at 22:41
1
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
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 at 23:46
1
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 at 11:14
 |Â
show 1 more comment
up vote
10
down vote
favorite
5
up vote
10
down vote
favorite
5
5
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?
algebraic-topology
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
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?
algebraic-topology
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
asked Jul 23 at 22:04
Naiche Cimarron Downey
542
542
2
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 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 at 22:41
1
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
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 at 23:46
1
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 at 11:14
 |Â
show 1 more comment
2
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 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 at 22:41
1
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
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 at 23:46
1
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 at 11:14
2
2
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 at 22:20
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 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 at 22:41
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 at 22:41
1
1
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
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 at 23:46
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 at 23:46
1
1
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 at 11:14
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 at 11:14
 |Â
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2
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 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 at 22:41
1
in these notes Adams claims that much of the motivation came from work of Serre.
– Andres Mejia
Jul 23 at 22:53
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 at 23:46
1
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 at 11:14