How to read homotopy schematics? 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.)
 A: 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.
A: 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:


*

*Definition. Definició 8.1.1, pages 207-208.

*Well-defined on homotopy classes. Proposició 8.1.2, page 208.

*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.
A: 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.   
