Timeline for The homology of the universal covering space, why so difficult to compute
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23 events
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| Aug 28, 2023 at 0:43 | comment | added | David Roberts♦ | The paper is since published: doi.org/10.4064/fm734-6-2020 (and the arXiv link is arxiv.org/abs/1807.06410) | |
| Mar 30, 2020 at 18:24 | comment | added | Phil Tosteson | Thanks for the elaboration Manuel. Morally, the failure of Koszul equivalence to agree with ordinary weak equivalence is the same as the failure of the Eilenberg--Moore spectral sequence to converge. For this reason, I doubt that you can use your construction to produce a spectral sequence converging to $H_*(\tilde X)$ whose $E_2$ page only depends $H_*(X)$ and $\pi_*(X)$. | |
| Mar 28, 2020 at 13:12 | comment | added | Manuel Rivera | @PhilTosteson right, in general, the (normalized) dg coalgebra of simplicial chains on a simplicial set $S$ is quasi-isomorphic ,but not weakly equivalent in the (stronger) Koszul sense, to the dg coalgebra of singular chains on the geometric realization $|S|$. However, if $S$ is a Kan complex then these two coalgebras are Koszul weakly equivalent. Also note that the Koszul weak equivalences and quasi-isomorphisms of coalgebras coincide for simply connected dg coalgebras (the spectral sequence behaves nicely in this case, since there are no arbitrary long monomials in degree $0$). | |
| Mar 27, 2020 at 22:09 | comment | added | Phil Tosteson | @GSM As an example of the sort of phenomenon I am thinking of. $C_*(S^1)$ is formal, hence quasi-isomorphic to $\mathbb Z[\epsilon]$ as an $E_\infty$ algebra. But the cobar construction applied to $\mathbb Z[ \epsilon]$ is $\mathbb Z[\mathbb N]$, not $\mathbb Z[\mathbb Z]$. | |
| Mar 27, 2020 at 21:54 | comment | added | Manuel Rivera | You are right, if you use the $E_{\infty}$-algebra of cochains, as in Mandell's approach, then yes, but if you use $E_{\infty}$-coalgebras you don't. However, there are still details to be worked out for the coalgebra version of Mandell's theorem. If you want to talk more you can email me! | |
| Mar 27, 2020 at 21:51 | comment | added | GSM | @ManuelRivera But if you use the Mandell Theorem, you need to assume some finiteness condition on homology groups? Or I'm saying something wrong ? | |
| Mar 27, 2020 at 21:43 | comment | added | Manuel Rivera | Depends on what you mean by "usual algebraic structure". I think the $E_{\infty}$-coalgebra of singular chains under the weak equivalces given by Koszul duality should determine the homotopy type of $X$, since by a theorem of Mandell, in the simply connected case (for example, for the universal cover) the homotopy type is determined by the $E_{\infty}$ structure under ordinary quasi-isomorphisms. | |
| Mar 27, 2020 at 20:50 | comment | added | GSM | @PhilTosteson As far as I understand $C_{\ast}(X)$ + its usual algebraic structure encodes $C_{\ast}(\tilde{X})$ + its usual algebraic structure + the fundamental group, but maybe not the homotopy type of $X$. | |
| Mar 27, 2020 at 19:00 | comment | added | Phil Tosteson | This is quite nice-- this parallels developments in Koszul duality theory due to Keller and others. However, I think there are subtleties that arise from your choice of weak equivalence. For instance, there should exist quasi-isomorphisms of $E_\infty$ coalgebras which do not determine equivalences of the cobar construction. That is, your choice of weak equivalence "knows about" more than just $C_*(X)$ with its usual algebraic structure. | |
| Mar 26, 2020 at 23:33 | history | edited | Manuel Rivera | CC BY-SA 4.0 |
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| Mar 26, 2020 at 13:31 | comment | added | Manuel Rivera | Technically the answer is not in the literature, but we are writing down the details and this has already been discussed in talks and private discussions! | |
| Mar 26, 2020 at 12:06 | comment | added | GSM | Thanks! So question 4 still open ? | |
| Mar 26, 2020 at 11:45 | comment | added | Manuel Rivera | @GSM Here is a slightly revised version of our paper: docs.google.com/… we will updated in the arXiv eventually. | |
| Mar 26, 2020 at 10:54 | comment | added | Manuel Rivera | @GSM yes $C_*(X) \otimes_{\tau} \mathbb{Z}[\pi_1(X)]$ is quasi-isomorphic to $C_*(\tilde{X})$ as en $E_{\infty}$-coalgebra. We are writing the details down but it is not so surprising based on the work of Benoit Freese which constructs a natural $E_{\infty}$-coagebra structure on the cobar construction $\Omega C$ of the underlying $A_{\infty}$-coalgebra of a $E_{\infty}$-coagebra $C$. | |
| Mar 26, 2020 at 10:18 | comment | added | GSM | I was reading your nice article (arXiv version) I found the answer to my precedent comment. Is the "question 4" still open ? | |
| Mar 26, 2020 at 9:04 | comment | added | GSM | Thank you! I was wondering if $C(X)\otimes_{\tau}\mathbb{Z}[\pi_{1}(X)]$ is quasi-isomorphic to $C(\tilde{X})$ as $E_{\infty}$-coalgebra ? | |
| Mar 26, 2020 at 9:02 | history | bounty ended | GSM | ||
| Mar 26, 2020 at 9:01 | vote | accept | GSM | ||
| Mar 26, 2020 at 3:51 | history | edited | Manuel Rivera | CC BY-SA 4.0 |
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| Mar 26, 2020 at 3:44 | history | edited | Manuel Rivera | CC BY-SA 4.0 |
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| Mar 26, 2020 at 3:27 | history | edited | Manuel Rivera | CC BY-SA 4.0 |
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| Mar 26, 2020 at 3:14 | history | edited | Manuel Rivera | CC BY-SA 4.0 |
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| Mar 26, 2020 at 3:08 | history | answered | Manuel Rivera | CC BY-SA 4.0 |