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David C
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As you know extending $P$-localization functors to non-nilpotent spaces is a very delicate matter. You can do it by homotopical localization techniques and. if you do it in this way you get functorial localization which work for non-nilpotent spaces "but" it is the aim of your question it is not "very explicit".

Now let me be very naive and let us go back to D. Sullivan's MIT notes "Geometric topology, localization, periodicity and Galois symmetry" chapter 2 (these notes are really beautiful to read). D. Sullivan first explains how to do $P$-localization when we have a CW-complex, of course he also explains that this CW-procedure gives a "nice localization" when the CW-complex is 1-connected (in that case it is our modern $P$-localization). He proceeds on the skeleton of the CW-complex, localizing cell by cell. Thus you can apply Sullivan's telescope construction to a circle and then to a wedge of circles and any CW-complex you want. The starting point being the localization of spheres through a telescope of maps $S^k\stackrel{\times p}{\rightarrow }S^k$ (you invert degree $p$-maps).

The point is that you can argue that this naive construction on non-nilpotent CW-complexes is not what you expected as a $P$-localization. D. Sullivan also explains how to do it when we have a Postnikov tower.

Let me add that there is another interesting way to $P$-localize which is due to C. Casacuberta and G. Peschke: "Localizing with respect to self-maps of the circle" Trans AMS 339 (1993) 117 – 140.

This $P$-localization is as "explicit" as a homological Bousfield's localization but the homotopy related to it is very interesting. In that setting, a space is $P$-local if and only if the $p$-power map on the loop space $\Omega X$ is a self homotopy equivalence if $p\notin P$.

As you know extending $P$-localization functors to non-nilpotent spaces is a very delicate matter. You can do it by homotopical localization techniques and in this way you get functorial localization which work for non-nilpotent spaces "but" it is the aim of your question it is not "very explicit".

Now let me be very naive and go back to D. Sullivan's MIT notes "Geometric topology, localization, periodicity and Galois symmetry" chapter 2. D. Sullivan first explains how to do $P$-localization when we have a CW-complex, of course he also explains that this CW-procedure gives a "nice localization" when the CW-complex is 1-connected (in that case it is our modern $P$-localization). He proceeds on the skeleton of the CW-complex, localizing cell by cell. Thus you can apply Sullivan's telescope construction to a circle and then to a wedge of circles and any CW-complex you want. The starting point being the localization of spheres through a telescope of maps $S^k\stackrel{\times p}{\rightarrow }S^k$ (you invert degree $p$-maps).

The point is that you can argue that this naive construction on non-nilpotent CW-complexes is not what you expected as a $P$-localization. D. Sullivan also explains how to do it when we have a Postnikov tower.

Let me add that there is another interesting way to $P$-localize which is due to C. Casacuberta and G. Peschke: "Localizing with respect to self-maps of the circle" Trans AMS 339 (1993) 117 – 140.

This $P$-localization is as "explicit" as a homological Bousfield's localization but the homotopy related to it is very interesting. In that setting, a space is $P$-local if and only if the $p$-power map on the loop space $\Omega X$ is a self homotopy equivalence if $p\notin P$.

As you know extending $P$-localization functors to non-nilpotent spaces is a very delicate matter. You can do it by homotopical localization techniques. if you do it in this way you get functorial localization which work for non-nilpotent spaces "but" it is the aim of your question it is not "very explicit".

Now let me be very naive and let us go back to D. Sullivan's MIT notes "Geometric topology, localization, periodicity and Galois symmetry" chapter 2 (these notes are really beautiful to read). D. Sullivan first explains how to do $P$-localization when we have a CW-complex, of course he also explains that this CW-procedure gives a "nice localization" when the CW-complex is 1-connected (in that case it is our modern $P$-localization). He proceeds on the skeleton of the CW-complex, localizing cell by cell. Thus you can apply Sullivan's telescope construction to a circle and then to a wedge of circles and any CW-complex you want. The starting point being the localization of spheres through a telescope of maps $S^k\stackrel{\times p}{\rightarrow }S^k$ (you invert degree $p$-maps).

The point is that you can argue that this naive construction on non-nilpotent CW-complexes is not what you expected as a $P$-localization. D. Sullivan also explains how to do it when we have a Postnikov tower.

Let me add that there is another interesting way to $P$-localize which is due to C. Casacuberta and G. Peschke: "Localizing with respect to self-maps of the circle" Trans AMS 339 (1993) 117 – 140.

This $P$-localization is as "explicit" as a homological Bousfield's localization but the homotopy related to it is very interesting. In that setting, a space is $P$-local if and only if the $p$-power map on the loop space $\Omega X$ is a self homotopy equivalence if $p\notin P$.

Source Link
David C
  • 9.9k
  • 3
  • 31
  • 58

As you know extending $P$-localization functors to non-nilpotent spaces is a very delicate matter. You can do it by homotopical localization techniques and in this way you get functorial localization which work for non-nilpotent spaces "but" it is the aim of your question it is not "very explicit".

Now let me be very naive and go back to D. Sullivan's MIT notes "Geometric topology, localization, periodicity and Galois symmetry" chapter 2. D. Sullivan first explains how to do $P$-localization when we have a CW-complex, of course he also explains that this CW-procedure gives a "nice localization" when the CW-complex is 1-connected (in that case it is our modern $P$-localization). He proceeds on the skeleton of the CW-complex, localizing cell by cell. Thus you can apply Sullivan's telescope construction to a circle and then to a wedge of circles and any CW-complex you want. The starting point being the localization of spheres through a telescope of maps $S^k\stackrel{\times p}{\rightarrow }S^k$ (you invert degree $p$-maps).

The point is that you can argue that this naive construction on non-nilpotent CW-complexes is not what you expected as a $P$-localization. D. Sullivan also explains how to do it when we have a Postnikov tower.

Let me add that there is another interesting way to $P$-localize which is due to C. Casacuberta and G. Peschke: "Localizing with respect to self-maps of the circle" Trans AMS 339 (1993) 117 – 140.

This $P$-localization is as "explicit" as a homological Bousfield's localization but the homotopy related to it is very interesting. In that setting, a space is $P$-local if and only if the $p$-power map on the loop space $\Omega X$ is a self homotopy equivalence if $p\notin P$.