User michael freedman - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:42:14Z http://mathoverflow.net/feeds/user/1643 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68952/geometry-of-complex-elliptic-curves Geometry of complex elliptic curves michael freedman 2011-06-27T18:34:07Z 2012-08-13T00:31:07Z <p>Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat? </p> http://mathoverflow.net/questions/98616/extension-of-surface-homeomorphism extension of surface homeomorphism michael freedman 2012-06-01T23:23:50Z 2012-06-12T21:07:18Z <p>Can anyone give me a reference (or proof sketch) for the fact that there are psuedo-Anosov diffeomorphisms of closed hyperbolic surfaces which do not extend over any handlebody? Thanks.</p> http://mathoverflow.net/questions/42629/can-all-n-manifolds-be-obtained-by-gluing-finitely-many-blocks/45635#45635 Answer by michael freedman for Can all n-manifolds be obtained by gluing finitely many blocks? michael freedman 2010-11-11T01:48:57Z 2010-11-11T21:29:04Z <p>I posted a paper on the arXiv, <a href="http://arxiv.org/abs/1011.2460" rel="nofollow">Group Width</a> which answers this question for manifolds of dim $>3$ with sufficiently complicated fundamental group (there will be no finite set of blocks). As Greg Kuperberg said, there are many interesting variations which remain open and are a nice challenge to technique, e.g. the case of simply connected manifolds.</p> http://mathoverflow.net/questions/30567/level-sets-of-morse-functions/45634#45634 Answer by michael freedman for Level sets of Morse functions michael freedman 2010-11-11T01:36:08Z 2010-11-11T21:26:26Z <p>I posted a short paper to the arXiv, <a href="http://arxiv.org/abs/1011.2460" rel="nofollow">Group width</a>, which answers this question at least for nonsimply connected manifolds. I think the same question for simply connected manifolds still deserves an answer.</p> http://mathoverflow.net/questions/42629/can-all-n-manifolds-be-obtained-by-gluing-finitely-many-blocks/44500#44500 Answer by michael freedman for Can all n-manifolds be obtained by gluing finitely many blocks? michael freedman 2010-11-02T01:05:24Z 2010-11-02T19:37:35Z <p>Thanks to Ian Agol for pointing out this question and a related one on levels of Morse functions - /305067/. In both case, for smooth manifold of dim $> 3$, as expected, there is no finite list of blocks ( or regular level components.) The idea is that one may define the "width" of a group, by representing the group G as the fundamental group of some complex K and then slicing K into "levels". The game to to arrange the slices so that the image of $\pi_1$ of each component of each level set maps a subgroup of small rank under the inclusion into $\pi_1(K)$. width( G) is defined as a Minmax over all slicings of all complexes K with $\pi_1 K = G$ of the rank of these image subgroups. I wrote a few pages to show that width( $\mathbb{Z}^k$) $= k-1$. The only slightly technical ingredient is Lusternick-Schnirelmann category. This answers negatively these finiteness questions since there are $d$ manifolds with $\pi_1 =\mathbb{Z}^k$ all $k$, as long as $d>3$. As soon as the notes are teXed, I can post them on the arxiv or math overflow.</p> http://mathoverflow.net/questions/36105/nonasymptotic-complexity-results nonasymptotic complexity results michael freedman 2010-08-19T18:10:19Z 2010-08-22T01:43:14Z <p>I recall hearing about a result, or maybe a cluster of results, in some area of complexity theory, probably algebraic, to the effect that there are known, specific, short formulas whose minimal derivation is known to be exceedingly long. Or perhaps it is a specific function that requires an exceedingly deep curcuit. The "philosophy" seemed to be that such examples would threaten to make the older asymptotic question obsolete: "Who cares about asymptotics if the constants are huge." Can anyone, given these hints, describe ( and direct me to) the result I overheard?</p> http://mathoverflow.net/questions/36108/geometry-of-null-homotopies geometry of null homotopies michael freedman 2010-08-19T18:40:53Z 2010-08-20T15:59:30Z <p>Given a homotopy class of map $f$ between unit spheres $S^n \to S^m, n>m$, let "stretch" be its "stretch factor" ( = inf over the homotopy class of the sup norm on the ( operator) norm on the first derivative). Suppose, as is usually the case, that some $k$th power of $f$ is null homotopic. For a fixed representative $f$, $$\textrm{stretch} (f^k) \leq k\cdot \textrm{stretch} (f).$$ Does anyone know an example where the stretching of the null homotopy $h$ for $f^k$ ( as a map from the $n+1$-disk, $h:D^{n+1}\to S^m$, $h_{|\partial D^{n+1}}=f^k$) must be "enormously" larger (or even just larger) than $k\cdot \textrm{stretch}(f)$? Since simply connected homotopy theory is decidable, the minimum stretch of the null homotopy $h$ is bounded by a recursive function of the stretch of the boundary condition, stretch($f^k$) - but this still allows scope for potentially extravagant examples. Are any known?</p> http://mathoverflow.net/questions/15519/topological-milnors-conjecture-on-torus-knots/16337#16337 Answer by michael freedman for topological "milnor's conjecture" on torus knots. michael freedman 2010-02-25T00:39:53Z 2010-02-25T00:39:53Z <p>Related to the early investigation of the Thom conjecture,the G-signature thm was used circa 1970 to give 4-ball genus bounds for torus knots which asyptocically (in some cases) were a fixed fraction of what we now know to be the smooth category answer. I belive Larry Tayor observed (in the '70s or early 80s) that these G-signature bounds hold in the topologically flat world as well. Thus, I believe, there there are families of torus knots where the the flat-4-ball geunus is known to be at least some known fraction of the smooth 4-ball genus. Sorry I don't have the references at hand.</p> http://mathoverflow.net/questions/4766/squares-in-stable-homotopy squares in stable homotopy michael freedman 2009-11-09T20:18:02Z 2009-11-10T22:12:33Z <p>I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b out. I'm wondering if there are any ( many) other exmples of this. What are the elements in the stable homotopy of spheres which a squares in the above sense?</p> http://mathoverflow.net/questions/36105/nonasymptotic-complexity-results/36313#36313 Comment by michael freedman michael freedman 2010-08-23T16:31:03Z 2010-08-23T16:31:03Z Thank you. I think this is what I had heard. Many of the other answers were quite interseting as well. Mike http://mathoverflow.net/questions/36108/geometry-of-null-homotopies Comment by michael freedman michael freedman 2010-08-20T21:14:23Z 2010-08-20T21:14:23Z Regarding Tom Goodwillie's question. My notation was poor: I should have written the kth power in the abelian group additively as kf not f^k. However as Tom correctly suggests it doesn't really matter: do some operation (add, iterate, smash, whatever) to define a null homotopic map and then ask if the null homotopy must be in some sense more complex than its initial condition. Maybe the fact that a 3 torus with its lie group fraiming needs a complicated ( index/8 odd)bounding framed 4-manifold tells us that the composed cube of the Hopf map, while it does die put up quite a struggle. Budney. http://mathoverflow.net/questions/36105/nonasymptotic-complexity-results Comment by michael freedman michael freedman 2010-08-20T00:05:50Z 2010-08-20T00:05:50Z Actually, I posted a second question a few minutes after this one, the second on homotopy-theory and geometry. Careful consideration will reveal that the two questions are nearly identical. This illistrates the impossiblility of defining computer science as seperate from mathematics. http://mathoverflow.net/questions/36105/nonasymptotic-complexity-results/36125#36125 Comment by michael freedman michael freedman 2010-08-19T23:56:50Z 2010-08-19T23:56:50Z Thank you for the reference, but the author only remindes the reader of Godel's incompleteness theorem. No example is produced with which requires a long proof ( within some system.) http://mathoverflow.net/questions/4766/squares-in-stable-homotopy/4905#4905 Comment by michael freedman michael freedman 2009-11-11T00:21:02Z 2009-11-11T00:21:02Z Thanks, that answer is quite helpful.