Timeline for Why are spectra indexed over the natural numbers?
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| Sep 9, 2010 at 13:37 | comment | added | Peter Arndt | But the relation between stable homotopy type and homology type is the other way round (where "homology" means ordinary, e.g. singular, homology): The homology functor comes from Top, goes via suspension spectrum into Spt, then from left to right through the whole chain on page 22, and in the end you take homology of the chain complex. At the first step, the linearizing, you lose information - so there are spaces with different stable homotopy type (i.e. non-isomorphic in Ho(Spt)) which become isomorphic in Ho(Spt(A)) and thus have the same homology. | |
| Sep 9, 2010 at 13:28 | comment | added | Peter Arndt | Whoops, that's a typo on the bottom of page 22! All the adjunctions right of Spt(A) induce equivalences of homotopy categories - the one on the left doesn't. Also the Z[-] should be below the arrow pair not on top. The fact that Ho(spectra of positive chain complexes) is equivalent to chain complexes the analogy that I thought would be helpful. | |
| Sep 9, 2010 at 9:10 | comment | added | roger123 | Thank you Peter. I was a bit sloppy and of course meant "shift down functor" instead of "shift functor". Ah, I can acquire a taste for your argument that it doesn't matter since one considers "the limit" only anyway. But I cannot follow Dundas on the bottom of page 22 where he says that the homotopy category of Spt is equivalent to the "linearized" one via the linearization functor. Aren't there spaces with the same stable homotopy type but different homology? | |
| Sep 9, 2010 at 0:28 | history | answered | Peter Arndt | CC BY-SA 2.5 |