I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not mention the term. Here's a question about the geometric meaning of the prism operator. Unfortunately, the answer it received did not include any geometric intuition, so that's my first request. My second question is what exactly is the cone construction? How does Hatcher circumvent it?

Summing up:

  1. Duplicate of this question: What is the geometric idea behind the prism operators?
  2. What exactly is the cone construction and how does Hatcher circumvent it?
  • $\begingroup$ I just mention that the EMS Tract vol 15 partly titled "Nonabelian algebraic topology" and advertised at pages.bangor.ac.uk/~mas010/nonab-a-t.html has a quite different approach to basic homology/homotopy by using cubical methods which are much more convenient for homotopies than simplicial methods. The theory using higher groupoids is quite intuitive, though harder to set up, but it does give stronger results. The Relative Hurewicz Theorem, in the strong form involving actions, becomes a consequence of a Seifert-van Kampen type theorem. $\endgroup$ – Ronnie Brown Jan 5 '15 at 20:59
  • $\begingroup$ Sounds awesome, but I doublt I'll understand anything :D $\endgroup$ – user153312 Jan 5 '15 at 21:41
  • $\begingroup$ There is a free pdf and a substantial introduction giving motivation and history. See also pages.bangor.ac.uk/~mas010/pdffiles/galway7.pdf for a recent presentation on how I was led to these ideas. $\endgroup$ – Ronnie Brown Jan 12 '15 at 22:34
  • $\begingroup$ @RonnieBrown from my low level, it looks like proving things like homotopy invariance would be far more pleasant using cubes than simplices. $\endgroup$ – user153312 Jan 12 '15 at 22:46
  • $\begingroup$ @RonnieBrown I like many of the things that appear on your page, e.g doing things without basepoints etc. Can you recommend a reading list and order to gently acquaint myself with your approaches? What are the prerequisites? $\endgroup$ – user153312 Jan 12 '15 at 22:56

@Exterior: The van Kampen theorem for a set of base points is dealt with in Topology and Groupoids, T&G, which also deals with covering spaces and orbit spaces from a groupoid viewpoint. A small correction to one part is here.

I was introduced to groupoids in 1965 by Philip Higgins whose downloadable book Categories and Groupoids is a very good account of the area. A proof of the most general van Kampen type theorem for a set of base points is given in this paper. The nice point is also that the proof, as given there by verifying the universal property, is as easy for groupoids as for groups, and also generalises to higher dimensions. Of course if you give the theorem for groupoids, you have to develop some of the algebra of groupoids, to explain the applications. If you want to see some current developments, try arXiv:1207.6404.

My first paper on this was published in 1967, but my books are still, and I am not sure why, the only topology texts in English to give the general theorem.

Writing the book now called T&G led me to ask in the 1960s about the potential use of groupoids in higher homotopy theory, and this led us after years of trying to a new realm.

I should also say that Massey's nice book "Singular homology" takes the cubical approach. The use of cubical sets with connections has brought cubical sets more into current use. I can give references on that if needed.


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