Timeline for Spaces of $n$-dimensional topological spaces whose fundamental group is given
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 6, 2021 at 15:15 | comment | added | user164898 | Better put a bound on the number of cells in $X$, or demand that the homotopy groups of $X$ are finitely generated, or something else along those lines. Otherwise you have a proper class of such spaces $X$ (this already happens when $n=2$ and $G$ is trivial), so no hope of such a moduli space. | |
| Nov 6, 2021 at 14:29 | comment | added | Connor Malin | Such a moduli space, if reasonably defined, would be a union of $B(\operatorname{haut}(X))$ over all $X$ with $\pi_1(X)=G$. Here $B$ denotes the classifying space of a monoid and $\operatorname{haut}$ denotes homotopy automorphisms. Spaces like this are useful, but not to study $\pi_1$. | |
| Nov 6, 2021 at 14:26 | history | edited | LSpice | CC BY-SA 4.0 |
Link to @YCor's comment
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| Nov 6, 2021 at 13:36 | history | edited | Abh | CC BY-SA 4.0 |
added 70 characters in body
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| Nov 6, 2021 at 13:33 | comment | added | YCor | You'll have no control, even on the homotopy type, if you don't assume path-connected, and you'll need even more (otherwise one can crook things by adding fake loops that are not paths). So "locally path-connected" would help too. Maybe a starting point if you want a moduli space is to figure out what you need when $G=1$. | |
| Nov 6, 2021 at 13:29 | history | asked | Abh | CC BY-SA 4.0 |