Is R^3 the square of some topological space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:57:12Z http://mathoverflow.net/feeds/question/60375 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space Is R^3 the square of some topological space? Richard Dore 2011-04-02T18:42:06Z 2013-01-19T00:12:41Z <p>The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \times X$ were homeomorphic to $\mathbb{R}$, then $X$ would be path connected. But then $X \times X$ minus a point would also be path connected. But $\mathbb{R}$ minus a point is not path connected.</p> <p>A next natural space to consider is $\mathbb{R}^3$. My intuition is that $\mathbb{R}^3$ also doesn't have a square root. And I'm guessing there's a nice algebraic topology proof. But that's not technology I'm much practiced with. And I don't trust my intuition too much for questions like this.</p> <p>So, is there a space $X$ so that $X \times X$ is homeomorphic to $\mathbb{R}^3$?</p> http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60378#60378 Answer by Tyler Lawson for Is R^3 the square of some topological space? Tyler Lawson 2011-04-02T19:10:50Z 2011-04-02T19:10:50Z <p>No such space exists. Even better, let's generalize your proof by converting information about path components into homology groups.</p> <p>For an open inclusion of spaces <code>$X \setminus \{p\} \subset X$</code> and a field $k$, we have isomorphisms (the relative Kunneth formula) <code>$$H_n(X \times X, X \times X \setminus \{(p,p)\}; k) \cong \bigoplus_{p+q=n} H_p(X,X \setminus \{p\};k) \otimes_k H_q(X, X \setminus \{p\};k).$$</code> If the product is $\mathbb{R}^3$, then the left-hand side is $k$ in degree 3 and zero otherwise, so something on the right-hand side must be nontrivial. However, if <code>$H_p(X, X \setminus \{p\};k)$</code> were nontrivial in degree $n$, then the left-hand side must be nontrivial in degree $2n$.</p> http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60389#60389 Answer by Henno Brandsma for Is R^3 the square of some topological space? Henno Brandsma 2011-04-02T21:25:00Z 2011-04-02T21:25:00Z <p><a href="http://blog.plover.com/math/R3-root.html" rel="nofollow">this blog post</a> refers to some papers with proofs. I've heard Robert Fokkink explain his proof and there he also told us the cohomological proof, which generalizes it to all Euclidean spaces of odd dimension.</p> http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/119296#119296 Answer by Adam Epstein for Is R^3 the square of some topological space? Adam Epstein 2013-01-19T00:12:41Z 2013-01-19T00:12:41Z <p>I didn't know that, but I did know this: we cannot have $S^2 = S\times S$ for any topological space $S$.</p>