Timeline for Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?
Current License: CC BY-SA 3.0
12 events
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| Nov 11, 2015 at 7:47 | comment | added | Sebastian Goette | @ViditNanda This is two years old by now. Did you make any progress? | |
| Aug 21, 2014 at 15:41 | history | edited | Vidit Nanda | CC BY-SA 3.0 |
Removed old update.
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| Jun 24, 2013 at 16:33 | comment | added | Ronnie Brown | The pair should be $(X,A)$ of course. | |
| Jun 24, 2013 at 9:50 | comment | added | Ronnie Brown | @Vidit: I've sent an email on this. One point is that a pair $(X,S)$ defines filtrations $E_n(X,A)$ which are a base point in dim $0$, $A$ in levels $1$ to $n-1$ and $X$ thereafter. So if $X$ already has a filtration $X_*$ then we get two filtrations for a given $n$. | |
| Jun 23, 2013 at 15:57 | comment | added | Vidit Nanda | Ronnie, I'm not sure I see an "obvious" bifiltration anywhere in this question. Could you explain why that would be the natural thing to do? I am happy to discuss this via email if things get too intricate for comment boxes. | |
| Jun 23, 2013 at 15:55 | history | edited | Vidit Nanda | CC BY-SA 3.0 |
Added a question
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| Jun 22, 2013 at 14:14 | comment | added | Ronnie Brown | @Vidit: Reading your question again, I would be inclined to go for defining invariants of bifiltered spaces, rather than invariants in spaces. Such bifiltered spaces in truncated form can be regarded as special cases of $n$-cubes of spaces, to which my work with Loday applies. I'm slowly writing up something on the bifiltered case, and my exposition [60] ``Triadic Van Kampen theorems and Hurewicz theorems'', Algebraic Topology, Proc. Int. Conf. March 1988, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57. (pdf on my web site) gives an idea. Also I missed the word "discrete"! | |
| Jun 22, 2013 at 10:53 | comment | added | Vidit Nanda | Thank you, Ronnie. I already have your book(s), being a fan if the groupoid point of view, so I will take a look. | |
| Jun 22, 2013 at 10:48 | comment | added | Ronnie Brown | @Vidit: You mention a "filtered cell complex". The book "Nonabelian algebraic topology" (2011) see my web page, pages.bangor.ac.uk/~mas010/nonab-a-t.html , builds algebraic topology directly from filtered spaces, via homotopically defined functors. Problem 16.1.17 is about relating these methods to Morse Theory. | |
| Jun 22, 2013 at 10:22 | comment | added | Vidit Nanda | Johannes: somewhere in between. I'd call it concretely-motivated curiosity. My thesis work involved simplifying a filtered cell complex via discrete Morse theory in a way that preserved algebraic topological invariants (persistent homology groups). As such, I find the process of constructing massive chain groups and then taking a huge quotient somewhat inefficient, and was wondering if there is a "compact representation" (both words used non-technically) of some object in the category of topological spaces itself from which one could derive the invariants whenever necessary. | |
| Jun 22, 2013 at 9:33 | comment | added | Johannes Ebert | Where did you meet this structure? In a concrete problem or just out of curiosity? | |
| Jun 22, 2013 at 0:21 | history | asked | Vidit Nanda | CC BY-SA 3.0 |