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I am attempting to start working through J.P. Mays A Concise Course in Algebraic Topology but can't seem to understand what it describes as "schematic indications" of how a given homotopy behaves on "the domain squares."

He first defines three path-functions: $$ F : x \to y, \qquad G : y \to z, \qquad H : z \to w. $$ and then notes that their compositions are associative -- I am sure there's a better way to put, he simply writes: $h \circ (g \circ f) \sim (h \circ g) \circ f$ -- this is also easy to believe since we "defined" composition of paths a bit earlier (by concatenating them and going "twice as fast", which I am more or less willing to accept for the moment).

He then defines a constant-loop path function $c(x) : x \to x$ and draws his picture of the "domain square". The top and bottom indicate the three paths we have defined, $f$, $g$, and $h$ -- and the left and the right indicate two of the identity functions, $c(x)$ and $c(w)$. He draws two slightly inclined line segments through the square, creative three volumes, which could now presumably be thought of as $f$, $g$ and $h$. Note that the bottom row doesn't say $f$-prime, it just says $f$; so I guess this I guess is where I am confused.

So I sort of get what this is saying -- these functions move points to other points in an 'orderly' way -- but I guess I am having trouble understanding what is intended by "domain square" and he does not really define it. The setup for some of this did reference metric spaces and some stuff about fundamental groups -- should that provide enough context to gather what the domain square is in this context and how to apply to decode these little schematic drawings? I would appreciate any help or clarification on these.

(In passing, and this is of course not necessary, if someone could suggest a slightly less, well, "concise" algebraic topology course book I would be grateful.)

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Something that might help: write down explicit formulas for the compositions $h \cdot (g \cdot f)$ and $(h \cdot g) \cdot f$ using the "twice as fast" method. Look at the way the domain $I$ is split up between the three individual paths ($f$, $g$, and $h$). Now look at just the top and bottom of the domain square. Can you see how each corresponds to a different composition/path? And how the slanted lines provide a continuous link between the two compositions/paths? –  Adam Saltz Aug 2 '11 at 3:23
    
So the line in between f and g is c-sub-y? That is helpful, thank you. –  Joseph Weissman Aug 2 '11 at 3:41
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IMHO May's book is a better as a reference than as a place to learn. If you want a similarly advanced book with more details spelled out, you might want to try Switzer's "Algebraic Topology: Homotopy and Homology". –  Aaron Mazel-Gee Aug 2 '11 at 4:45
    
By the way, the picture in question is on pg. 6 of the book, which you can view for free on May's website. –  Dylan Moreland Aug 2 '11 at 5:35
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I want to agree, in part, with Aaron. May's book is not a good place to first learn the material. He doesn't attempt to build your intuition or give you a geometric perspective on the constructions. What he does do is give a clear, categorical, modern perspective on the basics of algebraic topology. I highly recommend it as a second book, as it gives a very useful perspective, and lets you know understand some of the technical underpinnings in a no frills, no fuss way. Great results presented in a great way, if you're ready for them (and not JUST a reference) –  Aaron Aug 2 '11 at 6:57
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3 Answers

up vote 4 down vote accepted

You can certainly write this out. For example, we can describe the leftmost region (the "$f$-region") as $$ \{(s, t) \in [0, 1]^2 : t \geqq 4s - 1\}, $$ and here the homotopy is given by $$ (s, t) \mapsto f\biggl(\frac{4s}{t + 1}\biggr). $$ This jives with the picture: fixing a $t$, we traverse $f$ from $x$ to $y$ as $s$ goes from $0$ to $(t + 1)/4$, where we hit the boundary. You can do this sort of scaling for the other two regions and check that your functions agree on the vertical boundaries.

There's a slightly different presentation of this on pg 27 of Hatcher's book that you might like. He emphasizes that precomposing a path with a continuous map $\varphi\colon[0, 1] \to [0, 1]$ fixing $0$ and $1$ does not change the homotopy class.

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«I'm not sure that much is gained by writing out the equations for this homotopy» I am always very surprised with this point of view. –  Mariano Suárez-Alvarez Aug 2 '11 at 5:15
    
@Mariano Perhaps that's a bit too strong, but I think the picture is what stays with you. I'll try to make this more, um, positive. –  Dylan Moreland Aug 2 '11 at 5:18
    
This was enormously helpful -- and thank you so much for suggesting the Hatcher, it is very good. –  Joseph Weissman Aug 2 '11 at 23:22
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Maybe these notes could help you. They're written in Catalan, but mathematical formulas are language-free I guess. :-) (Though, if you have problems with it, just let me know and I'll provide translations for what you need.)

Properties of the product of paths are stated, drawn and proved in:

  1. Definition. Definició 8.1.1, pages 207-208.
  2. Well-defined on homotopy classes. Proposició 8.1.2, page 208.
  3. Associativity, units, inverses. Proposició 8.1.3, pages 209-211.

Other advanced books on Algebraic Topology, less "concise" may be: Spanier, Hatcher and Massey. I would use the last one as my first book on the subject, together with the second. The first one and May's I would use them as unavoidable references.

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I fixed your links. You should replace the .co.uk by .com and it's better to get rid of all the noise in the amazon links except for http://www.amazon.com/dp/#some number#/. The reason why the links break is explained here. –  t.b. Aug 2 '11 at 9:52
    
Ok. Thanks, Teo. –  a.r. Aug 2 '11 at 10:56
    
No problem at all. By the way: are you this Agustí? I really like that title :) –  t.b. Aug 2 '11 at 11:14
    
Yes. Are you talking about my PhD thesis title? -Maybe a litle bit presumptuous. :-) –  a.r. Aug 2 '11 at 12:44
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A great old-fashioned algebraic topology book I love that I wish more people would take a look at is C.F.Maunder's ALGEBRAIC TOPOLOGY. It has very modern notation and yet is very geometric at the same time,with historical notes.I'm itching to try and teach my first algebraic topology course out of both it and May. A very comprehensive-and CHEAP!-algebraic topology course! –  Mathemagician1234 Aug 17 '11 at 3:52
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Writing out the formulas works nicely if the homotopy is fairly simple. Otherwise,it's not really gonna help. The book I found-and am still finding-most helpful for this is John McCleary's A First Course In Topology. I heartily recommend it to most students as a beautifully written first brush with topology from a visual and historical point of view. I'd go so far to say it contains the absolute minimum amount of topology needed before entering graduate school. Trying to learn algebraic topology from May is like trying to teach yourself anatomy by reading a few medical journals. What book would work best for you to learn from depends on whether you prefer a geometric or an algebraic approach to the subject. For the former, there's Hatcher. For the more abstract,functorial approach, there's either May, Spainer or the gorgeous book by Tammo tom Dieck. Which do I prefer? Neither. I learned the most from books and notes that took a balanced approach to the subject-and sadly,there aren't many. The best book I've ever seen in this regard is Joseph Rotman's An Introduction To Algebraic Topology. Like everything by Rotman, it's totally modern, deep and incredibly clear-yet it has many geometric discussions and references to the literature. THAT'S the one I'd use if I were you and use May as a supplemental text. If you could do that,you'd pretty much be ready for anything a topology qualifier could throw at you.

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...Joe is the OP; Patrick merely edited. –  J. M. Aug 3 '11 at 4:16
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I stand corrected-my bad. –  Mathemagician1234 Aug 5 '11 at 16:38
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