I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty basic stuff, but it's really cool to me that there are such clear topological analogues to the usual exact sequences in homological algebra. Still, I can't even get a clear picture of what a smash product looks like for any but the most basic of spaces, and based cones/suspensions/loopspaces make my head hurt.

a) Will I be alright if in my head I just sort think of a smash product as a usual product (for example), with the understanding that I need to tack on an extra condition that I really shouldn't think too hard about?

b) Why all the fuss about based homotopy theory, anyways?

c) While I'm at it, can anyone suggest a book that is less terse? I feel like this one rarely gives the motivation and visual intuition that I'd like...

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    $\begingroup$ For c), have you looked at chapter 4 of Hatcher's book? Everything there is very geometric. $\endgroup$ Feb 15 '10 at 9:44
  • $\begingroup$ I read Hatcher Ch. 1-3 but then departed from it to read other things -- I'll definitely check it out again. Thanks for the suggestion. $\endgroup$ Feb 17 '10 at 10:05
  • $\begingroup$ My favorite topology texts are ELEMENTS OF COMBINATORIAL AND DIFFERENTIAL TOPOLOGY and ELEMENTS OF HOMOLOGY THEORY,both by the outstanding Russian topologist V.V.Pravalov. I find both of them to give just the right balance of rigor and geometric intution. To me,most of the standard texts go too far in either direction. $\endgroup$ Jul 7 '10 at 2:58

Another book with pictures of reduced suspension etc. is Ronnie Brown's Topology and Groupoids. see http://www.bangor.ac.uk/~mas010/topgpds.html. Which is also excellent for non-based stuff.

Don't believe all you hear about the unbased case being grotty! It is beautiful, but it is possibly easier to learn Alg. Top. in the based situation first, especially if it is that the someone else has decided you should do. :-)

I have put some material that might help (Abstract Homotopy...) on my n-Lab home page (follow the links from Tim Porter (found by a search)).

For a) I had something like your problem when I started, but then thought of the based cylinder as a cylinder with a long base point! Of course, you really need to squidge that line to a point. It is safe when mapping out of a smash to retain the subspace that is to be squidged just always mapping all of it to the base. (In other words, don't agonise about the smash at this stage. Use it as a device for the moment and after you learn to use it and see how it behaves its strangeness will probably have dissapated.)

  • $\begingroup$ Wow, that's great that you've got notes on a lot of what I'm struggling with! I'm accepting this answer for that reason (among others). $\endgroup$ Feb 17 '10 at 10:10
  • $\begingroup$ @Aaron It depends on what you will need in future research, but the Menagerie notes on my n-Lab personal homepage amy also be useful later on. I tend to reuse material, once it is Latexed it is simpler to use diagrams etc that already exist. (Of course, that may mean that I stay blocked at the stage of my understanding from when it was first written! For that I need help and questions, queries and info on typos are very useful for that.) $\endgroup$
    – Tim Porter
    Feb 17 '10 at 11:20

I've got answers for b.) and c.).

b.) The reason based homotopy is so important is that in unbased homotopy theory, there's no good notion of a homotopy group. Now you may say, "what about the fundamental groupoid?", but that is somewhat misleading. To actually deal with the higher homotopy groupoids, you in fact need higher category theory, and all of the complicated notions of equivalence that come with it. So if you're trying to do classical homotopy theory, you need basing to deal with the higher homotopy data.

c.) I really like May's book, but if you want a good geometric picture as well, you might want to look through Spanier or Hatcher as a "companion" book. You may also want to look through Switzer or Whitehead, but they are much less focused on the geometric aspects of Algebraic Topology.

  • $\begingroup$ As a defender of fundamental groupoid, let me say that the treatment of something as simple as showing that the fundamental group of a circle is infinite cyclic in based theory is awkward, whilst the version that goes via a fundamental groupoid on two base point is more geometric, it avoids the use of complex analysis ideas yet retains the essence of them. It does not confuse methodology keeping things algebraic and topological. $\endgroup$
    – Tim Porter
    Feb 15 '10 at 14:11
  • $\begingroup$ Absolutely. I'm talking about the higher homotopy groupoids. At least, I have no idea how to describe them without higher category theory. I mean, you could use some kind of simplicial construction, but that's the only way I know how to deal with all of it. Then again, you're the expert, so I defer to your judgement. $\endgroup$ Feb 15 '10 at 14:37
  • $\begingroup$ In dimension one higher than the fundamental groupoid, Hardie, Kamps and Kieboom have a very neat construction. It can be used to gain an idea about non-based constructions yet does not involve using higher category theory. Ronnie Brown and Phil Higgins use a multibased approach rather than a single base point and that works well. With regard to the original question, that filtered space version of homotopy theory does lead to sensible constructions without the smash that can cause problems as Aaron finds. $\endgroup$
    – Tim Porter
    Feb 15 '10 at 16:51
  • $\begingroup$ That's cool! However, surely you'd agree that the motivation for based constructions was the difficulty of dealing with higher homotopy groupoids, no? $\endgroup$ Feb 15 '10 at 18:18
  • $\begingroup$ I think it is more historical. This is possibly discussed in Ronnie's online article 'From groups to groupoids'. There is the point that may of the early 'groups', we actually presented more like groupoids, basically since there was what we would call a group ACTING on a set, and that leads naturally to a groupoid, but once groupoids had gone below the horizon, people probably wanted to get this nice group out rather than a groupoid but the groupoid is very useful even if handling the fundamental group. (see Ronnie's book for this.) $\endgroup$
    – Tim Porter
    Feb 15 '10 at 19:59

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