Timeline for What would be the simplest analog of Langlands in algebraic topology?
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14 events
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| Aug 26, 2018 at 18:08 | comment | added | მამუკა ჯიბლაძე | @Watson No idea. Do you have any specific circumstances in mind? | |
| Aug 26, 2018 at 15:54 | comment | added | Watson | Would arithmetic topology be useful to formulate such an analogue? | |
| Aug 23, 2018 at 19:52 | comment | added | მამუკა ჯიბლაძე | @skd Thank you for bringing this up! I more and more think that I should start with a question about class field theory first. I believe the analog in algebraic topology of what you describe in the second part of your comment is actually a combination of some Poincaré-like duality (relating homology and cohomology) with something like connecting homomorphism for the exponential short exact sequence, relating cohomology with integer coefficients and shifted-by-one cohomology with circle group coefficients. | |
| Aug 23, 2018 at 19:21 | comment | added | skd | ... admits a deep relationship with the theory of modular forms (via the $\mathbf{E}_\infty$-ring of topological modular forms, and its numerous variants). Moreover, I've heard some people say that there should be a geometric interpretation of TMF cocycles in terms of "2-vector bundles", but I don't know what these are (this paper seems relevant: arxiv.org/abs/1805.04146). tl;dr: the discussion in the link Drew provided probably isn't the simplest analogue of Langlands, but it definitely seems to be the "right" one. | |
| Aug 23, 2018 at 19:17 | comment | added | skd | ...there is an isomorphism $\hat{\mathrm{H}}^{-2}(G; \mathbf{Z}) \to \hat{\mathrm{H}}^0(G; L^\times)$; the left hand side is $G_\mathrm{ab} = \mathrm{H}_1(G; \mathbf{Z})$, and the right hand side is $K^\times/\mathrm{N}(L^\times)$. This defines the local Artin homomorphism. Also: the link Drew pointed to is very relevant. I don't know anything about Langlands, but as far as I know (from high-level discussions with people) the n=2 case of Langlands is very closely related to the theory of modular forms. It should be no surprise that chromatic homotopy theory at height 2 also... | |
| Aug 23, 2018 at 19:11 | comment | added | skd | The cohomological approach to class field theory is discussed in Milne's notes jmilne.org/math/CourseNotes/CFT.pdf; the essential idea is that $\mathrm{Gal}(K^\mathrm{ab}/K)$ can be thought of $\mathrm{H}_1(\mathrm{Gal}(K^\mathrm{sep}/K; \mathbf{Z})$, where $K$ is a nonarchimedean local field. In any case, I would say that Tate's theorem (II.3.11 in Milne) is the "most important" part of the proof, not this result. This is because Tate's theorem implies that if $L/K$ is a finite extension with Galois group $G$, then... | |
| Aug 23, 2018 at 15:32 | comment | added | მამუკა ჯიბლაძე | @DrewHeard Many thanks for sharing this! It is very interesting, although WAY more sophisticated than what I am asking here. | |
| Aug 23, 2018 at 7:41 | comment | added | Drew Heard | Semi-related: mathoverflow.net/questions/7283/topological-langlands | |
| Aug 23, 2018 at 6:36 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
not sure if this is the terminology used...
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| Aug 23, 2018 at 5:26 | comment | added | მამუკა ჯიბლაძე | @skd Well even that I am not sufficiently familiar with, and it would be great if you could put some detail in a sort of partial answer | |
| Aug 23, 2018 at 5:12 | comment | added | skd | Just a remark (which you probably already know about): "abelianization of the fundamental group is the first homology group, as some remote relative of class field theory" is a statement used in the cohomological approach to class field theory. | |
| Aug 23, 2018 at 5:00 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
added 56 characters in body
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| Aug 23, 2018 at 4:48 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
added 122 characters in body
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| Aug 23, 2018 at 4:43 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 4.0 |