Timeline for Homology theory constructed in a homotopy-invariant way
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2020 at 15:18 | history | edited | Ronnie Brown | CC BY-SA 4.0 |
added comments and links
|
| Oct 27, 2020 at 15:01 | history | edited | Ronnie Brown | CC BY-SA 4.0 |
added comments and links
|
| Oct 27, 2020 at 14:55 | history | edited | Ronnie Brown | CC BY-SA 4.0 |
added comments and links
|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Apr 28, 2014 at 10:50 | comment | added | Ronnie Brown | @Johannes: It looks like I need to write an expository article on, say, "Crossed complexes and singular homology". There are two aspects of the latter: what does it do, and what is it? I suppose my answer was more relevant to the former rather than the latter. Also one needs to look at the history of homology theory: the book "History of Topology" Edited I.M. James, is relevant (although the word "groupoid" does not appear there). | |
| Apr 25, 2014 at 14:53 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
typo
|
| Apr 24, 2014 at 20:24 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
further point on "formal sums"
|
| Apr 24, 2014 at 19:56 | review | Low quality posts | |||
| Apr 24, 2014 at 20:08 | |||||
| Apr 24, 2014 at 17:53 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
added some more explanation
|
| Apr 24, 2014 at 9:54 | comment | added | Ronnie Brown | From a fundamental crossed complex $\Pi X_*$ of a filtered space one obtains a fundamental group(oid) and "homology" modules $\pi_n(\Pi X_*,x)$ at $x \in X_0$. If $X_*$ is the skeletal filtration of a CW complex then these are the homology modules of the universal covers at various base points. These ideas are due to JHC Whitehead (1949). However to prove some new facts about $\Pi$ as a functor one uses an equivalent "cubical higher homotopy groupoid" $\rho(X_*)$. This approach replaces the usual "formal sums" by actual homotopical compositions. Not a short story! | |
| Apr 23, 2014 at 22:52 | comment | added | Johannes Hahn | In fact: Even if you goal was to describe another homology theory, I cannot see anything like that in your post. How does a "higher homotopy groupoid" give a homology theory in any usual sense of the word? Shouldn't there be a bunch of abelian groups (or at least a series of objects from an abelian category) somewhere? | |
| Apr 23, 2014 at 22:49 | comment | added | Johannes Hahn | The question is specifically about a homotopical version of singular homology not just any homology. If you think that your post indeed constitutes an answer to Guillaume's question, then perhaps you should add a paragraph detailing how the constructions that you mentioned can be used to retrieve the singular homology (for sufficiently nice spaces if you think that it cannot be done for all spaces). Maybe it's just me and my lack of knowledge about homotopy theory, but I cannot see something like that in your post. | |
| Apr 23, 2014 at 16:36 | comment | added | Ronnie Brown | @Johannes: Maybe it can't be done for spaces. Grothendieck in Section 5 of "Esquisses d'um Programme" argues that topological spaces are inadequate for (his kind of!) geometry. We try to explain what can be done for filtered spaces, which do often arise geometrically, e.g. as CW-complexes. As in my first paper, 1963, we are looking for categories which are "adequate and convenient for all purposes of topology". That is the point which needs discussion, and evaluation. The key question is "purposes". What does one expect of a "homology theory", and on what should it be defined? | |
| Apr 23, 2014 at 15:55 | comment | added | Ronnie Brown | @Johannes: Thanks for opening up to discussion. The question asked for homotopically defined homology. I try to explain that this can be done for filtered spaces, using an old construction of Blakers, 1948, involving fundamental group(oid)s and relative homotopy groups, which we have called $\Pi$. So it is homotopy invariant, on filtered spaces. Baues calls it a crossed chain complex. The question is how to compute it, and that is solved by higher van Kampen theorems. The usual procedure is to start with a space and then produce a singular set; whose realisation has a filtration! | |
| Apr 23, 2014 at 13:33 | comment | added | Johannes Hahn | A wild guess: Your post has been voted down because nothing in it even resembles an answer to Guillaume's question. It just seems as if you have copy-pasted your standard text about groupoids, higher van-kampen theorems etc. without even considering that it might have little or nothing to do with the question. | |
| Apr 23, 2014 at 9:34 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
added a link to a related mathoveflow question
|
| Apr 21, 2014 at 19:44 | comment | added | Ronnie Brown | I am intrigued by the fact that my answer has been marked down. However I have never heard any objection to the proofs in the works cited. I would be happy to discuss the matter in private or in public. It is true that the idea of Higher Homotopy Seifert-van Kampen Theorems is not mentioned in any text on algebraic topology except my own, although the first 2-d theorem was published in 1978. | |
| Sep 29, 2012 at 21:18 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
some additional references and refinements of the argument
|
| Jul 24, 2012 at 20:51 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
added another link for Whitehead's theorem on free crossed modules
|
| Jan 27, 2012 at 17:43 | history | answered | Ronnie Brown | CC BY-SA 3.0 |