Wikipedia:Reference desk/Archives/Mathematics/2013 August 1
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August 1
editBrouwer's Fixed Point Theorem
editIn our article on Brouwer fixed-point theorem under the proof by homology section, I had two questions regarding the content there. The first, it says that the function F is a retraction, but I don't see why the boundary points would have to be fixed; I don't see that that is crucial to the proof though. Second, it says that the case for n > 2 is more difficult and requires homology, and while that proof makes sense, I don't understand why we can't use the same idea since πn(Sn) = Z for n > 0, unless the closed unit ball has a non-trivial group for higher n. Any help would be greatly appreciated:-)Phoenixia1177 (talk) 06:20, 1 August 2013 (UTC)
- Every boundary point is fixed for F, that follows directly from the definition of F: if x is a boundary point, then whatever f(x) is (provided by definition, that f(x) ≠ x), the ray from f(x) through x meets the boundary at x, thus F(x) = x. --CiaPan (talk) 07:42, 1 August 2013 (UTC)
- Wow, I feel really really stupid now. I flipped the direction in my mind as x going through f(x) to the boundary! I had a feeling I was getting something wrong since that's not the first time I've seen that version of the proof. At any rate, is there a similarly simple reason why we can't use the higher homotopy groups?Phoenixia1177 (talk) 07:54, 1 August 2013 (UTC)
- Same 'direction flip' problem was explained at the article's talk page here in 2006. Anyway I do not understand the connection between a retraction and a group homomorphism, either, so I'm unable to help you with the central step of the proof. --CiaPan (talk) 08:52, 1 August 2013 (UTC)
- As for why not use "the same idea" in higher dimensions, I'd say that the homology approach is the same idea. In the dimension 1 case, since the fundamental group is abelian, the fundamental group is the same as the dimension 1 homology group. So when you want to generalize to higher dimensions you can choose to generalize to higher homotopy or to higher homology. But homology is simpler and suffices to make the argument work, so using the homotopy groups is unneccesary. Staecker (talk) 13:43, 1 August 2013 (UTC)
- That makes sense; the wording of going from fundamental group to homology makes it sound like there is a reason we couldn't use the higher homoptopy groups, this is what was throwing me for a loop (ha!). I'm guessing the reason the n = 2 case uses the fundamental group is because historically, and in order learned (usually?), the fundamental group would come first?Phoenixia1177 (talk) 13:59, 1 August 2013 (UTC)
- That's what I'd assume. In my experience as a student and teacher, people generally learn fundamental group, then homology groups, then (if at all) homotopy groups. Most working mathematicians know basic things about the fundamental group and homology groups. Most nontopologists (and some topologists) know very little about the higher homotopy groups. Staecker (talk) 16:59, 1 August 2013 (UTC)
- Speaking of what working mathematicians know and homology, brings me to another question. If you grabbed a mathematician, at random, would they be familiar with all the various cohomology theories out there? In general, I've always been curious, how many mathematicians have a working knowledge of various topics (like Galois Cohomology, or K-Theory, or Model Categories, etc.). Algebraic Topology and Algebraic Geometry, to me, seems to become "a nightmare in the key of category theory" after a point, so I've always wondered just how common a detailed understanding of it is.Phoenixia1177 (talk) 04:20, 2 August 2013 (UTC)
- Anecdotal: I have math Ph.D., and I know very little about Galois Cohomology (loved classic Galois theory, but that's as far as I went with algebra). In contrast to Staecker's general path, I learned homotopy groups before homology groups, via a fun course in knot theory. Anyway, my impression is that working pure mathematicians work in their own "silos", and only have cursory knowledge once outside of their expertise domain. In grad school, when I was studying Ito calculus, some of my peers were doing coxeter groups and dynkin diagrams. We had fun chatting about eachother's topics, but I doubt any of us went on to learn the non-central-to-them topics in great depth. To sum up my impression, if you take a random pure math prof in the USA, and ask about fields in which she does not publish, she will probably top out around the level of a second year grad student. SemanticMantis (talk) 15:33, 5 August 2013 (UTC)
- Thank you:-) I study math for my own enjoyment and have, for a long time, had this impression (because I read every textbook/paper in the same author "voice") that every mathematician knows most everything. Once this moved up into an actual conscious thought, I realized that made little sense, and was curious what the actual situation is.Phoenixia1177 (talk) 04:17, 6 August 2013 (UTC)
- Anecdotal: I have math Ph.D., and I know very little about Galois Cohomology (loved classic Galois theory, but that's as far as I went with algebra). In contrast to Staecker's general path, I learned homotopy groups before homology groups, via a fun course in knot theory. Anyway, my impression is that working pure mathematicians work in their own "silos", and only have cursory knowledge once outside of their expertise domain. In grad school, when I was studying Ito calculus, some of my peers were doing coxeter groups and dynkin diagrams. We had fun chatting about eachother's topics, but I doubt any of us went on to learn the non-central-to-them topics in great depth. To sum up my impression, if you take a random pure math prof in the USA, and ask about fields in which she does not publish, she will probably top out around the level of a second year grad student. SemanticMantis (talk) 15:33, 5 August 2013 (UTC)
- Speaking of what working mathematicians know and homology, brings me to another question. If you grabbed a mathematician, at random, would they be familiar with all the various cohomology theories out there? In general, I've always been curious, how many mathematicians have a working knowledge of various topics (like Galois Cohomology, or K-Theory, or Model Categories, etc.). Algebraic Topology and Algebraic Geometry, to me, seems to become "a nightmare in the key of category theory" after a point, so I've always wondered just how common a detailed understanding of it is.Phoenixia1177 (talk) 04:20, 2 August 2013 (UTC)
- That's what I'd assume. In my experience as a student and teacher, people generally learn fundamental group, then homology groups, then (if at all) homotopy groups. Most working mathematicians know basic things about the fundamental group and homology groups. Most nontopologists (and some topologists) know very little about the higher homotopy groups. Staecker (talk) 16:59, 1 August 2013 (UTC)
- That makes sense; the wording of going from fundamental group to homology makes it sound like there is a reason we couldn't use the higher homoptopy groups, this is what was throwing me for a loop (ha!). I'm guessing the reason the n = 2 case uses the fundamental group is because historically, and in order learned (usually?), the fundamental group would come first?Phoenixia1177 (talk) 13:59, 1 August 2013 (UTC)