# Freudenthal suspension theorem plus latex question.

I was looking at this paper - are there any easier references that go through the proof of Freudenthal suspension theorem? I need to use this to prove something about spheres. My lecturer was saying there is a special case that only deals with spheres.

Also, I was wondering, how does the author LaTeX

$$(f+g)(x_1,x_2,x_3,\ldots,x_n)= \begin{cases} f(2x_1,x_2,\ldots,x_n)&\text{if } 0 \leq x_1 \leq \frac{1}{2},\\ g(1-2x_1,x_2,\ldots,x_n)& \text{if }\frac{1}{2} \leq x_1 \leq 1 \end{cases}$$

so that it looks like the document.

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@t.b. Thanks, but that doesn't work in texworks. So I'm sort of stuck. –  simplicity Apr 7 '12 at 15:37
You need to use \usepackage{amsmath} in the preamble (before \begin{document}) I'm pretty sure texworks ships with amstex, so that's the only modification you need has that included. Here's the documentation. –  t.b. Apr 7 '12 at 15:40
Thanks t.b. that actually solved it. –  simplicity Apr 7 '12 at 15:41
DO NOT WRITE "...". Write "\ldots". They won't look the same in the document; the former won't have proper spacing. –  Michael Hardy Apr 7 '12 at 15:56
LaTeX questions should be asked at tex.stackexchange.com. –  Qiaochu Yuan Apr 7 '12 at 15:58

This question seems to be asking two completely different questions: The first is a LaTeX formatting question (which is not entirely appropriate here), and the second is asking for an "easier" source for the Freudenthal Suspension Theorem.

As far as proofs of this theorem go, this one looks pretty easy. The standard way to prove it these days is using spectral sequences, and I have a feeling you don't want that. Although they make the argument more streamlined, there is a considerable amount of effort behind understanding the spectral sequence framework. The theorem seems to hinge on their Proposition 5.2, the Homotopy Excision Property. This is the part that I've seen proven using spectral sequences, but maybe the proof in your source is closer to Freudenthal's original proof (which was from the 30's, after all). For a better reference I was going to recommend Hatcher, but since it's the only citation in the source you gave I assume you've already look at Hatcher, correct? ;)

You only say "I need to use this to prove something about spheres," but it might help if I knew what you need to prove about spheres. Furthermore, when your lecturer says "there is a special case that only deals with spheres," is he referring to the stable homotopy groups of spheres? The theorem stated in the pdf you referenced is about $(n-1)$-connected CW complexes, but $S^n$ is certainly one of those spaces.

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$$(f+g)(x_1,x_2,x_3,\ldots,x_n) = \begin{cases} f(2x_1,x_2,\ldots,x_n), &\text{if }0 \leq x_1 \leq \frac{1}{2},\\ g(1-2x_1,x_2,\ldots,x_n), &\text{if }\frac{1}{2} \leq x_1 \leq 1 \end{cases}$$

Do not write "...", as suggested in a comment above; instead write \ldots.

Here is a philosophy professor's "writing guide" for papers his students turn in: http://spot.colorado.edu/~huemer/writing.htm. Note this point:

Ellipses: use spaces.

Bad: “The past consistency...calls for some explanation...”

Ok: “The past consistency . . . calls for some explanation. . . .”

$\TeX$ and $\LaTeX$ are sophisticated and this proper usage is built-in to them.

I normally use \ldots between commas and \cdots between binary operators or binary relations, thus:

$$(x_1,\ldots,x_n)$$ $$x_1+\cdots+x_n$$ $$x_1<\cdots<x_n$$

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Why don't you use &hellip; in the html part of your post and code blocks to display your LaTeX code, then? That's not okay either :) –  t.b. Apr 7 '12 at 16:18