visualizing what's going on in based homotopy theory, et al. - MathOverflow

visualizing what's going on in based homotopy theory, et al.

2010-02-15T09:26:30Z

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...

Answer by Harry Gindi for visualizing what's going on in based homotopy theory, et al.

2010-02-15T09:56:20Z

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.

Answer by Tim Porter for visualizing what's going on in based homotopy theory, et al.

2010-02-15T10:31:44Z

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.)