Andy Putman
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 Feb3 comment Calculus Problems - Analysis MO is intended for questions at the mathematics PhD level and above. I've voted to close. Dec16 awarded Caucus Aug16 comment Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions @semiclassical : please don't send questions like this to MO, where it will be quickly closed. Jun21 comment examples of polyclic groups @noah s: So you don't consider zero-dimensional manifolds to be manifolds? Jun2 answered Transitive action of $\text{SL}(n,\mathbb Z)$ May31 awarded Yearling May26 awarded Editor May26 revised Homology subgroups generated by non-intersecting cycles added 183 characters in body May26 comment lifting a product of commutators of standard generators on 2-manifolds @FilipParker : Fix a piecewise-linear structure on your surface and homotope the curve such that it is piecewise-linear (this is trivial; just fix a bunch of points on it and "pull them tight"). Then jiggle the vertices a little so that all of the line segments in your curve intersect transversely. May26 answered Homology subgroups generated by non-intersecting cycles May12 awarded Commentator May12 comment lifting a product of commutators of standard generators on 2-manifolds @FilipParker : My last comment constitutes a complete proof. If you don't understand it, I recommend meditating on the part of any book on algebraic topology that discusses changing the base point in the fundamental group. I remember Massey's book having a particularly nice treatment of this. May11 comment lifting a product of commutators of standard generators on 2-manifolds @FilipParker : The homotopy lifting property tells you that that a loop $f$ lifts to a closed curve if and only if any curve which is homotopic to $f$ while fixing the base point lifts to a closed curve. Freely homotoping the curve lets the base point move, and thus corresponds to conjugating $f$. Apr26 comment lifting a product of commutators of standard generators on 2-manifolds (continued) The second is that the surface is of this form. This can be derived from the classification of surfaces. I recommend perusing Section 1.3 of Farb-Margalit's "Primer on mapping class groups" for the details. 3. You are correct that these are typos; thanks for pointing them out! Apr26 comment lifting a product of commutators of standard generators on 2-manifolds @FilipParker : 1. The surjection depends on $x$; for example, if you are dealing with $\alpha_1^7 \beta_1$ then you choose the homomorphism that takes $[\alpha_i]$ to $0$ for all $i$, that takes $[\beta_i]$ to $0$ for all $i \neq 1$, and that takes $[\beta_1]$ to $1$. 2. There are 2 points here. The first is that this is a presentation for $\pi_1$ of a punctured surface, which is discussed in any book that deals with surfaces and the fundamental group (e.g. Massey's book on the fundamental group). (continued) Apr25 answered lifting a product of commutators of standard generators on 2-manifolds Apr24 comment why say complex multiplication of elliptic curves is beautiful You are wrong about the definition of an abelian extension. The class group is (by its very definition) always an abelian group. The word "abelian" in "abelian extension" refers to the Galois group of the extension. Mar19 comment Books (and supporting material) that are useful in deconstructing one's intuition? I could immediately see that there must be a solution. Here's how. It's clear that you can connect A to A and C to C by paths that don't touch. Now imagine "cutting" the square open along those paths. Since those cuts stop in the middle of the paper (i.e. at the positions of the top A and top C), they don't actually end up cutting the paper into two pieces (to do so, you'd have to finish cutting to the edge of the paper!). So there must exist a path in the "cut open paper" from B to B. Tape the cuts back together and you've got your solution. Dec19 comment Higher dimensional Euclidean geometry problem MO is intended for questions at the mathematics PhD student level and above. I've voted to close. Sep4 comment Best Books to learn Proof-Based Linear Algebra and Matrices MO is intended for topics at the graduate school level and above.