Timeline for Why the Dold-Thom theorem?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2016 at 14:57 | comment | added | Jesse C. McKeown | The $SP$ construction is (we have to admit) simpler; it's easy enough to describe its approximants $SP^{(n)}(X) = (X^n) /_{strict} (\Sigma_n)$ and their inclusions, while building up enough continuous strict algebra to really nail down the exercises may be more daunting. Incidentally, an over-emphasis on $\mathbb{Z}[-]$ will give misleading ideas about the correct maps between these things. There are $E^\infty$-maps $\mathbb{Z}[\sph^2]\to\mathbb{Z}[\sph^n]$ that are nontrivial, and therefore not strict abelian! | |
| Mar 13, 2016 at 11:35 | comment | added | Bruno Stonek | Ah, this seems to be a typical problem of infinite CW complexes resolved just by working in compactly generated spaces... | |
| Mar 13, 2016 at 11:26 | comment | added | Bruno Stonek | @მამუკაჯიბლაძე I'm having a hard time with the German, but Dold-Thom's Satz 6.10.III seems to say that the inclusion $SP(X)\to \mathbb{Z}[X]$ is a weak homotopy equivalence for $X$ a "countable simplicial complex". I'm guessing that one can replace the "simplicial" hypothesis by CW, but what about "countable"? Also, what is the reason behind the fact that most expositions on the Dold-Thom theorem focus on the SP construction instead of the $\mathbb{Z}[-]$ one? The way I see it, the one with $\mathbb{Z}$ is better because we don't need X to be connected... | |
| Oct 23, 2014 at 19:07 | vote | accept | Chris Gerig | ||
| Oct 23, 2014 at 18:55 | history | edited | Chris Gerig | CC BY-SA 3.0 |
Consolidating the important comments
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| Oct 8, 2014 at 4:31 | comment | added | მამუკა ჯიბლაძე | @JesseC.McKeown commenting to your first comment :D Indeed the crux is the fact that abelianizing turns cofibration sequences into fibration sequences and so connects homology of the source with homotopy of the target. This also works for other homology theories, giving things like $\tilde E_*X\cong\pi_*(E\wedge X)$ (although one has to stabilize $X$ into $\Sigma^\infty X$, and weaken "abelian" to "triangulated"...) | |
| Oct 7, 2014 at 22:11 | comment | added | Jesse C. McKeown | There is (or ought to be) a slogan: "homology is abelianized homotopy"; the earliest inkling of this was Poincaré's formula $H_1 \simeq (\pi_1 / [\pi_1,\pi_1])$. And so compare/contrast (as I'm sure you have) $SP^\infty X \simeq Z[X]$ and $J_\infty X \simeq \Omega \Sigma X$; but that's not close enough to being an argument. | |
| Oct 7, 2014 at 22:06 | comment | added | Jesse C. McKeown | Hm. I think the kicker in the whole thing is Exercise 3, which is basically that Abelian Groups are an Abelian Category (that should be intuitive!) --- what's novel is that this part still works if our Abelian groups have interesting (but not too wild) topology. Or I could say "trust me, for homotopy theory, this is pretty darn intuitive", but that wouldn't be nice. | |
| Oct 7, 2014 at 19:09 | history | answered | Jesse C. McKeown | CC BY-SA 3.0 |