Timeline for Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?
Current License: CC BY-SA 4.0
12 events
when toggle format | what | by | license | comment | |
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Dec 5, 2021 at 5:44 | comment | added | Arshak Aivazian | @AchimKrause Thanks a lot! | |
Nov 25, 2021 at 12:42 | comment | added | Achim Krause | Hatcher, for example. The idea is to obstruct the existence of such a map by looking at the effect on cohomology rings. | |
Nov 25, 2021 at 4:36 | comment | added | Arshak Aivazian | @AchimKrause Still, I fail to understand why there is no continuous mapping from $\mathbb{R}P^3$ to $\mathbb{R}P^2$ inducing an isomorphism of fundamental groups. Can you tell me which textbook to look at? | |
Nov 10, 2021 at 21:48 | comment | added | Achim Krause | Here's a counterexample to that version as well. Write $(BC_p)^n$ for the n-skeleton of the standard CW structure on $BC_p$. Let $p, q$ be two different primes. Then $(BC_p)^2 \times (BC_q)^4$ and $(BC_p)^4 \times (BC_q)^2$ have the same fundamental group, but there's no map either way that induces an isomorphism on it. | |
Nov 10, 2021 at 21:44 | comment | added | Achim Krause | That's a weird thing to expect, don't you think? That there's always a map in at least one direction, but you don't know beforehand in which? | |
Nov 10, 2021 at 20:34 | comment | added | Arshak Aivazian | @HenrikRüping I beg your pardon, of course, I meant there is a mapping in some direction inducing an isomorphism | |
Nov 10, 2021 at 7:31 | comment | added | HenrikRüping | It is also a counterexample to (3). There is no mapping from $\mathbb{R}P^3$ to $\mathbb{R}P^2$ inducing an isomorphism on $\pi_1$. | |
Nov 10, 2021 at 7:01 | comment | added | Arshak Aivazian | Thanks! I have added a 3rd question that matches the title exactly, maybe less trivial. | |
Nov 10, 2021 at 6:45 | vote | accept | Arshak Aivazian | ||
Nov 10, 2021 at 6:45 | comment | added | Achim Krause | You can replace $\mathbb{R}P^\infty$ by $\mathbb{R}P^3$. | |
Nov 10, 2021 at 6:41 | comment | added | Arshak Aivazian | Thanks a lot! I need to think a little about your answer. Are there any counterexamples among finite-dimensional complexes? | |
Nov 10, 2021 at 6:39 | history | answered | Achim Krause | CC BY-SA 4.0 |