Timeline for Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?
Current License: CC BY-SA 4.0
23 events
when toggle format | what | by | license | comment | |
---|---|---|---|---|---|
Mar 16 at 13:11 | comment | added | Dmitri Pavlov | @D.R.: Yes, see Sections 15–24 in my notes: dmitripavlov.org/notes/topology.pdf | |
Mar 16 at 5:34 | comment | added | D.R. | Hi, did you ever come across something along these lines/write something along these lines that you liked? | |
Nov 10, 2018 at 18:25 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 54 characters in body
|
Nov 10, 2018 at 18:13 | comment | added | Denis Nardin | @DmitriPavlov I still think that Goerss-Jardine come closest of all available references, even if they are slanted towards homotopy and need the reader to bring their own motivation and examples. At least they provide a logically self-contained treatment of the topic. Unfortunately there is a dearth of introductory books in homotopy theory written from a simplicial perspective | |
Nov 10, 2018 at 18:11 | comment | added | Vincenzo Zaccaro | Yes, you are right. Unfortunally, I don't belive that there exists a such "complete" reference. Anyway, If I remember well also some Milnor's paper could be interesting to you. | |
Nov 10, 2018 at 18:08 | comment | added | Dmitri Pavlov | @DylanWilson: I am not sure why this reference would do "enough topology to 'present' surfaces with small simplicial sets associated to a cell structure". No topological spaces need to be mentioned at all in such a source. One can draw pictures of small simplicial sets directly without any reference to topological spaces. There are older sources that do this for abstract simplicial complexes, so the question is whether something similar exists for simplicial sets, which are more convenient. | |
Nov 10, 2018 at 18:05 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 105 characters in body
|
Nov 10, 2018 at 18:02 | comment | added | Dylan Wilson | I'm having trouble imagining how something could exist that answers your questions with all these restrictions in place. You want some reference that only does things with simplicial sets, but then does enough topology to 'present' surfaces with small simplicial sets associated to a cell structure (not there singular simplicial set) and then computes homology using the normalized chain complex... but surely any such reference would just use the cellular chain complex (which will agree with the normalized thing from the associated simplicial set) and bypass simplicial sets entirely, right? | |
Nov 10, 2018 at 18:01 | comment | added | Dmitri Pavlov | I am aware of Greg Friedman's article, it is very good, but the only place in it where homology of simplicial sets is mentioned is Example 6.3, which takes 7 lines of text and suffers from the same 2 problems that I pointed out in the main post: it does not use normalization of chains and does not compute any examples. | |
Nov 10, 2018 at 17:56 | comment | added | Vincenzo Zaccaro | Try to take a look at An elementary illustrated introduction to simplicial sets of Greg Friedman. It could be very good for your purposes. It misses some explicit computation but it is very clear! | |
Nov 10, 2018 at 17:51 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 253 characters in body
|
Nov 10, 2018 at 17:44 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 253 characters in body
|
Nov 10, 2018 at 17:38 | comment | added | Dmitri Pavlov | Goerss and Jardine explicitly assume familiarity with singular homology in this fragment: "Recall that the integral singular homology groups H^*(X; Z) of the space X are defined to be the homology groups of the chain complex ZSX." So it could not possibly be an answer to the first version of the question either. | |
Nov 10, 2018 at 17:29 | comment | added | Dmitri Pavlov | @DenisNardin: Yes, this is what the question highlighted in bold asks for: "an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology". | |
Nov 10, 2018 at 17:25 | comment | added | Najib Idrissi | @DmitriPavlov Homology is defined on page 5. Then Section 2.2 deals with normalized complexes etc. It's not explicitly about homology of spaces, rather about simplicial abelian groups, but implicitly you can take $A = \mathbb{Z}X$ throughout. | |
Nov 10, 2018 at 17:23 | comment | added | Dmitri Pavlov | I edited the question to reiterate the point about normalized chains and simple examples. | |
Nov 10, 2018 at 17:22 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 168 characters in body
|
Nov 10, 2018 at 17:19 | comment | added | Dmitri Pavlov | @DenisNardin: I was aware of these 5-6 lines in Goerss-Jardine before I wrote this post, but how could they possibly be an answer to my question? Goerss and Jardine write "Recall that the integral singular homology groups H^*(X; Z) of the space X are defined to be the homology groups of the chain complex ZSX." and assume prior knowledge of singular homology in this tiny fragment. Normalization of chains is not used, no examples are computed, and they clearly assume prior knowledge of algebraic topology, including homology. | |
Nov 10, 2018 at 17:12 | comment | added | Dmitri Pavlov | @Qfwfq: While Section I.4 in Gelfand–Manin indeed contains the basic definitions (which is better than nothing), their definition of simplicial chains does not use normalized chains, which makes it unsuitable for computations with simple examples. They also compute exactly zero examples of homology. | |
Nov 10, 2018 at 17:02 | comment | added | Dmitri Pavlov | I am sorry, would you mind pointing me to the exact section in Goerss-Jardine that treats homology of simplicial sets? I cannot find anything close. | |
Nov 10, 2018 at 14:14 | comment | added | Qfwfq | What about Gelfand--Manin's Homological algebra? (Maybe it's not an in-depth treatment as other references specifically addressed towards simplicial sets though) | |
Nov 10, 2018 at 13:20 | comment | added | Najib Idrissi | There is the book of Goerss and Jardine, Simplicial Homotopy Theory. There are some remarks on singular homology, but they aren't necessary for the definitions: everything is purely in simplicial terms. | |
Nov 10, 2018 at 0:38 | history | asked | Dmitri Pavlov | CC BY-SA 4.0 |