Misprint in Switzer's Algebraic Topology? I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can be what he states.
Let 
$\beta':S^1 \rightarrow I \vee S^1$ be defined by $(2t,_\ast)$ if $t \leq 1/2$ and $(\ast,2t-1)$ if $t > 1/2$. Consider the quotient map $q:I \rightarrow S^1$ given by $q(t) = e^{2\pi t}$. Switzer then claims that the composition 
$\alpha= (q \vee 1) \circ \beta': S^1 \rightarrow S^1 \vee S^1$ is given by $\alpha(t) = (4t,_\ast)$ it $t \leq 1/4$, $\alpha(t) = (\ast,2t-1/2)$ if $1/4 \leq t \leq 3/4$ and $\alpha(t) = (4(1-t),\ast)$ if $3/4 \leq t \leq 1$. However, I get that the composition is $(4s,\ast)$ for $t \leq 1/2$ and $(\ast, 2t-1)$ for $t \geq 1/2$. Is Switzer wrong, or am I misunderstnading something?  
 A: $\beta'$ in Switzer's book is different. Look at the general definition, there is one more variable. For a circle embedded as $x^2 + y^2 = 1$, it works like that: emphasize points $(1, 0)$ and $(0, 1)$, they are $S^0$ over which $S^1$ is suspended. The $t$ of Switzer's book in $\beta'(s, x, t)$ is more or less vertical coordinate in the plane (zero when $y = -1$, increasing twice slower than $y$ so $t = 1$ means $y = 1$). So you "fold" lower half of the circle into the segment $I$ and then map upper half to $S^1$. Using single real coordinate on $S^1$ the map $\beta'\colon S^1 \to I\vee S^1$ sends $[0, 1/4]$ and $[3/4, 1]$ to $I$, $[1/4, 3/4]$ to $S^1$.
With this in mind, it should be possible to write down $\beta'$ as a formula of $t$. Though it's not necessary: things happening in that chapter in Switzer's book are not so complicated to imagine geometrically. It is way easier and more pleasant than manipulating with formulas, and you wouldn't spend your time dealing with meaningless errors like forgetting which semicircle is parametrized by $[0, 1/2]$ and which by $[1/4, 3/4]$.
Generally, I would not recommend using explicit formulaic parametrizations for circles and segments like Switzer does in the book unless the situation is too complicated to be imagined in detail. Hard to understand, easy to make mistakes, and no room for geometric intuition. I would suggest reading about higher homotopy groups in some other source, for example in (freely available online) book by A. Hatcher: it has beautiful (meaningful!) pictures and explains what's going on conceptually, not just by writing messy formulas.
