Timeline for A generalization of the Borsuk Ulam theorem

8 events

when toggle format	what		by	license	comment
Mar 24, 2017 at 8:27	comment	added	Ali Taghavi		@lvan thank you. What about the same question for $\mathbb{R}^3 \to \mathbb{R}^2$.?
Mar 22, 2017 at 14:29	comment	added	Ivan Izmestiev		There is no such a map. Let $p$ be an interior point of $f(\mathbb{R}^2)$ (if there is no interior point, then $f$ is constant). The restriction of $f$ to $\mathbb{R}^2 \setminus f^{-1}(p)$ is a continuous map onto a disconnected set. Hence $\mathbb{R}^2 \setminus f^{-1}(p)$ is disconnected, in particular $f^{-1}(p)$ is infinite (even uncountable).
Mar 22, 2017 at 13:40	comment	added	Ali Taghavi		Thank you again for your very interesting answer. It seems that the powerfull method of characteristic classes is more applicable in compact cases for example $\mathbb{R}P^4$ rather than non compact bases. As an example of non compact case, I am wondering if this question is an elementary question: "Is there a continuous map $f:\mathbb{R}^2 \to \mathbb{R}$ such that every level set is a finite set?
Mar 21, 2017 at 20:14	history	edited	Michael Albanese	CC BY-SA 3.0	Inserted letters with accents.
Mar 21, 2017 at 17:45	history	edited	Ivan Izmestiev	CC BY-SA 3.0	added 7 characters in body
Mar 21, 2017 at 15:33	history	edited	Ivan Izmestiev	CC BY-SA 3.0	added example of non-orientable manifold (first paragraph)
Mar 21, 2017 at 14:14	vote	accept	Ali Taghavi
Mar 21, 2017 at 13:54	history	answered	Ivan Izmestiev	CC BY-SA 3.0