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I am currently reading Singular Homology Theory and Cohomology on my own mainly from Hatcher's "Algebriac Topology" and "Topology and Geometry" by Bredon. Quite often it happens that it takes a lot of time to understand the motivation behind certain concepts and sometimes I even get confused that whatever I have understood is even correct or am I just interpreting things in a wrong way in order to simply them for my own understanding.

Now I was reading this https://mathoverflow.net/questions/28268/do-you-read-the-masters question on MO where a lot of people have said that reading the original research papers on certain topics have cleared many of their misconceptions and held their understanding on a firm footing.

So I was wondering that Singular Homology and Cohomology Theory, even though whatever I am studying have been discovered a long time ago, but are there any $\bf{original \ research \ papers/surveys}$ which discuss them from the beginning (i think that the term "beginning" is somewhat vague, but i hope it conveys the point which I am trying to make).

Thanks

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    $\begingroup$ I guess you'd want to go back to Poincare if you want to understand the beginning of homology theory. Betti numbers were the first invariant introduced and, I believe, it wasn't until later that Emmy Noether realised that they were part of the more intricate group structure that we associate with homology nowadays. $\endgroup$
    – Dan Rust
    Dec 7 '14 at 13:03
  • $\begingroup$ Of course if you want to go back to the very basic motivation of homology, it all started out with the Euler characteristic, independently discovered by Descartes and Euler for triangulations of the sphere, and then extended to higher dimensional spheres by Schläfli (proved by Poincare) - references can be found here (many not in english, but the Coxeter references look interesting). $\endgroup$
    – Dan Rust
    Dec 7 '14 at 13:13
  • $\begingroup$ @DanielRust Thanks for the reference. Do you have any reference for the paper of Emmy Noether mentioned above ? $\endgroup$
    – wanderer
    Dec 7 '14 at 13:30
  • $\begingroup$ Unfortunately I think Noether's contribution was less formal than a published document. Works that directly followed her observations are contained within the references of her wikipedia artcile (specifically her contributions to topology) here. "Noether mentions her own topology ideas only as an aside in one 1926 publication." $\endgroup$
    – Dan Rust
    Dec 7 '14 at 13:35
  • $\begingroup$ Motivated by your question, I found this writeup of a lecture given by Peter Hilton on the history of homotopy and homology theory. It's terse in way of details, and lacking references, but I think it gives a great overview of the historical development of the involved ideas. $\endgroup$
    – Dan Rust
    Dec 7 '14 at 14:34
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I am not surprised that you find this a difficult area to get into! One has to some extent to appreciate the historical background, which is nicely covered in the books "History of Topology" Edited IM James, particularly the article by Lefschetz, and also in the book by Dieudonné.

The early articles on topology used terms like cycles modulo boundary, but were not too clear on what these meant, it seems! Then Poincaré produced his famous articles and in particular the notion of "formal sums of oriented simplices" and this led to the formula $\partial \partial =0$ which we know and love. The singular theory, using maps of a simplex to a space, had difficulty at first as the orientation led to chain groups with elements of order $2$, for degenerate simplices. This problem was brilliantly solved by Eilenberg with the notion of ordered simplices, and so the formula $\partial = \sum_i (-1)^i \partial _i$. His paper on "Singular homology theory" is well worth reading.

The notion of "formal sum" came from integration theory where is was natural to write

$$\int _C f + \int_D f \quad \text{as} \quad \int_{C+D} f. $$ So we needed free abelian groups, etc, as shown by Emmy Noether.

To give an answer to your question, what to read of the masters, it would be interesting to know your reaction to the book "Foundations of Algebraic Topology" by Eilenberg and Steenrod, but to start with their exposition of examples of theories satisfying their axioms.

You should not assume that basic algebraic topology is in its final, canonical, state. I have been plugging a new line in the book advertised here, to which there is an Introduction in a recent presentation. This book is quite advanced, assuming some familiarity with notions of fundamental group(oid) and homotopy. My web page for the book has a free pdf, and the book has introductory chapters giving the main intuitions. To set up properly this approach does require some sophisticated work! I'd be interested to know if this helps.

The point is that to my mind the hazy part of traditional algebraic topology is the border between homology and homotopy, and that is what this new approach reworks. It is based on the idea of using cubical methods to form actual compositions, generalising ideas for the fundamental group(oid), rather than "formal sums". Note that compositions of simplices are difficult to deal with. See this mathoverflow discussion of cubical v. simplicial.

Another paper you should look at is this by Mike Barr, 1995. It also refers to an old topology book by Seifert and Threlfall.

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  • $\begingroup$ thanks a lot for referring "History of Topology". I just saw the table of contents and it seems to match my requirements. Earlier I planned to read "Foundations..." but I got too busy with Hatcher/Bredon. I'll definitely have a look at it. The article is also very nice (though I have just read first few pages only). I think I'll have to consult your book "Topology and Groupoids" to learn further in that area. $\endgroup$
    – wanderer
    Dec 7 '14 at 19:28

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