User johndoe - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:44:47Z http://mathoverflow.net/feeds/user/25577 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/127922#127922 Answer by johndoe for Giving $Top(X,Y)$ an appropriate topology johndoe 2013-04-18T06:11:24Z 2013-04-19T09:24:04Z <p>It might be of interest to the original poster to know that $Top(X,Y)$ endowed with the compact-open topology guarantees at least one direction in the implication. In other words, if $F_*\colon X\times\mathbb{I}\to Y$ is continuous then $F\colon \mathbb{I}\to Top(X,Y)$ is. Hence you can safely interpret any homotopy as a path in the function space $Top(X,Y)$, but (allegedly) there are paths in $Top(X,Y)$ which do not correspond to homotopies.</p> <p>I, for my part, would like to see a counterexample for the opposite direction, since all the counterexamples I know of seem to use a space different than $\mathbb{I}$.</p> <p>Reference: Dugundji, Topology, Chapter XII, Theorem 3.1.</p> http://mathoverflow.net/questions/104198/classify-mathbbrn-bundles/104203#104203 Answer by johndoe for Classify $\mathbb{R}^n$-bundles johndoe 2012-08-07T14:33:22Z 2012-08-07T18:45:26Z <p>I assume that by $\mathbb{R}^n$-fiber bundles you mean differentiable bundles with fiber diffeomorphic to $\mathbb{R}^n$. So you are forgetting the linear structure of $\mathbb{R}^n$, keeping only the smooth structure. It is a fact that $\textit{Diff}(\mathbb{R}^n)$ deformation retracts to $\textit{GL}(n,\mathbb{R})$, hence yes, the classification of rank $n$ real vector bundles is essentially the same as that of $\mathbb{R}^n$-fiber bundles. I don't know whether this holds at the topological level already (that is, whether $\textit{Homeo}(\mathbb{R}^n)$ deformation retracts to $\textit{GL}(n,\mathbb{R})$).</p> <p>EDIT: it seems that at the topological level the two beasts are different, at least for some $n$. The result is contained in a paper by William Browder entitled "Open and closed disc bundles", Ann. of Math. (2) 83 (1966), 218-230. It is freely available on JSTOR. As a consequence of the results in that paper, there exists some $n$ for which $\textit{Homeo}(\mathbb{R}^n)$ does not deformation retract onto $\textit{GL}(n,\mathbb{R})$. As far as I know, $n>2$.</p> http://mathoverflow.net/questions/133865/does-the-gluing-procedure-in-robert-walds-book-general-relativity-yield-a-haus Comment by johndoe johndoe 2013-06-16T10:05:25Z 2013-06-16T10:05:25Z have you tried looking at the equivalent condition that the quotient map be open and $\sim$ be a closed subset of $\times^2\bigsqcup_{U\in\mathcal{U}}\tilde{M}_U$? http://mathoverflow.net/questions/133700/every-real-holomorphic-hamiltonian-vector-field-on-a-kahler-manifold-is-killing Comment by johndoe johndoe 2013-06-14T09:15:19Z 2013-06-14T09:15:19Z @Yemon Choi: yes. http://mathoverflow.net/questions/133724/existence-of-geodesics-in-continuous-metrics Comment by johndoe johndoe 2013-06-14T09:11:49Z 2013-06-14T09:11:49Z this paper seems relevant: <a href="http://arxiv.org/abs/1212.6962" rel="nofollow">arxiv.org/abs/1212.6962</a> http://mathoverflow.net/questions/130472/monotone-homotopy/130503#130503 Comment by johndoe johndoe 2013-05-13T17:48:12Z 2013-05-13T17:48:12Z I don't understand. How is &quot;Directed Homotopy Theory&quot; related to the OP question? Thanks for any clue. http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/128362#128362 Comment by johndoe johndoe 2013-04-23T12:51:53Z 2013-04-23T12:51:53Z I am interested in the reference to Lawvere's paper, thank you in advance. http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology Comment by johndoe johndoe 2013-04-18T05:53:16Z 2013-04-18T05:53:16Z @Amr: it seems you misplaced some * in the edited version http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/127853#127853 Comment by johndoe johndoe 2013-04-17T17:08:14Z 2013-04-17T17:08:14Z doesn't Isbell consider the case where I (as in the OP question) might run over the whole category of spaces? In other words, it seems to me that the OP question is a special case of the problem considered in Isbell's survey and might very likely have an affirmative answer. Am I wrong? http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology Comment by johndoe johndoe 2013-04-17T17:03:14Z 2013-04-17T17:03:14Z to Anton Fetisov: why is the case I rather useless? http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/127847#127847 Comment by johndoe johndoe 2013-04-17T14:10:12Z 2013-04-17T14:10:12Z shouldn't A be the interval I in order to prove the inexistence of such a topology? http://mathoverflow.net/questions/121620/why-does-gln-have-no-spinor-representations Comment by johndoe johndoe 2013-02-12T17:20:05Z 2013-02-12T17:20:05Z i believe the missing word here is faithful. take a look at lemma 5.23 in lawson-michelsohn. http://mathoverflow.net/questions/121168/group-of-diffeomorphisms-of-a-manifold Comment by johndoe johndoe 2013-02-08T10:50:13Z 2013-02-08T10:50:13Z you might want to take a look at banyaga's book &quot;the structure of classical diffeomorphism groups&quot;, 1997. http://mathoverflow.net/questions/118955/real-symmetric-matrix-has-at-least-one-real-eigenvalue-an-elementary-algebraic Comment by johndoe johndoe 2013-01-15T10:16:08Z 2013-01-15T10:16:08Z this question seems a duplicate of this one <a href="http://mathoverflow.net/questions/118626/real-symmetric-matrix-has-real-eigenvalues-elementary-proof" rel="nofollow" title="real symmetric matrix has real eigenvalues elementary proof">mathoverflow.net/questions/118626/&hellip;</a> (and the OP has actually contributed to it by answering) http://mathoverflow.net/questions/116888/on-john-von-neumann-and-quantum-mechanics/116981#116981 Comment by johndoe johndoe 2012-12-21T18:27:43Z 2012-12-21T18:27:43Z too much alcohol during festivities? http://mathoverflow.net/questions/108560/what-is-a-good-book-on-topological-groups Comment by johndoe johndoe 2012-10-01T17:23:58Z 2012-10-01T17:23:58Z You might want to take a look at Hewitt-Ross (two volumes). http://mathoverflow.net/questions/107163/a-function-that-is-defined-everywhere-but-has-unknown-values Comment by johndoe johndoe 2012-09-14T13:06:45Z 2012-09-14T13:06:45Z Consider the function that is 1 everywhere if at least one proposed example suits your tastes, and 0 everywhere if whichever suggestion comes you can always find some reason to reject it.