Timeline for What are the uses of the homotopy groups of spheres?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2012 at 5:11 | comment | added | Spice the Bird | I think that TMF and TAF may provide a relationship between homotopy groups of spheres and special values of L-functions. | |
| Apr 28, 2010 at 17:52 | comment | added | Tyler Lawson | As you can see in the very nice Wikipedia article that Pete linked to, the group of differentiable structures on the n-sphere has a cyclic subgroup, but the cokernel injects into the quotient of the n'th stable stem by the image of the J-homomorphism (it is usually isomorphic). For example, the group of differentiable structures on S^9 has Z/2 x Z/2 as a quotient. | |
| Apr 28, 2010 at 17:33 | comment | added | Qfwfq | "For instance, as someone else commented on the related question, if one understood even the stable homotopy groups of spheres very well, one would therefore have a near complete understanding of the group (I assume that n=4 ) of differentiable structures on the n -sphere". Wasn't the set of differentiable structures on a given $n-$sphere just a cyclic group under connected sum? What is there to be understood about the structure of that group? | |
| Apr 28, 2010 at 16:20 | comment | added | Pete L. Clark | @Dev: Thanks for your comment; I changed the language slightly to something which I hope is more apt. By the way, I agree with you that special values of L-functions are a reasonable number-theoretic analogue of homotopy groups of spheres. | |
| Apr 28, 2010 at 16:19 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Apr 28, 2010 at 15:35 | comment | added | Dev Sinha | "I wonder whether in practice, the homotopy groups of spheres are the application, not the tool." I would say this is fairly on the mark, though I wouldn't use the word "application." Homotopy groups of spheres in part serve as a measuring stick for our understanding of homotopy theory. For example, one way to measure the depth of the construction of elliptic cohomology was the consequence of its existence for homotopy groups of spheres. I once heard a number theorest talk about special values of L-functions similarly, as a good place to measure progress. | |
| Apr 28, 2010 at 12:24 | history | answered | Pete L. Clark | CC BY-SA 2.5 |