A basic construction in homotopy is Puppe sequences. Given a map $A \stackrel{f}{\to} X$, its homotopy cofiber is the map $X\to X/A=X \cup_f CA$ from $X$ to the mapping cone of $f$. If we then take the cofiber again, something remarkable happens: $(X/A)/X$ is naturally homotopy equivalent to the suspension $\Sigma A$ of $A$. This isn't hard to see geometrically; a nice picture and discussion can be found in pages 397-8 of Hatcher. If we iterate this, we end up with a sequence $$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma(X/A) \to \Sigma^2 A \to \cdots$$
in which each map is the homotopy cofiber of the previous map. If we then apply a functor which sends cofiber sequences to exact sequences, we get a long exact sequence. This can be understood as the origin of long exact sequences of cofibrations in (co)homology, using the fact that $H^n(X)=H^{n+1}(\Sigma X)$.
One subtlety of this construction is that under the natural identifications of $(X/A)/X$ and $((X/A)/X)/(X/A)$ with $\Sigma A$ and $\Sigma X$, the map $\Sigma A\to\Sigma X$ is not the suspension of the original map $f$, but rather its negative (i.e., $-1\wedge f: S^1\wedge A=\Sigma A \to \Sigma X=S^1\wedge X$, where -1 is a map of degree -1). The geometric explanation for this can neatly be seen in Hatcher's picture, where the cones are successively added on opposite sides, so the suspension dimensions are going in opposite directions.
However, you usually don't need to worry about this sign issue. First, since a map of degree -1 is a self-homotopy equivalence (even homeomorphism) of $\Sigma A$, we could just change our identification of $(X/A)/X$ with $\Sigma A$ by such a map and then we would just have $\Sigma f:\Sigma A \to \Sigma X$ (note though that then we are not changing how we identify the next space in the sequence with $\Sigma X$, which breaks some of the symmetry of the picture). Alternatively, if we only care about the Puppe sequence because of the long exact sequences it gives us, we could note that an exact sequence remains exact if you change the sign of one of its maps.
My question is: is there any situation where these signs really do matter and have interesting consequences? Might they be somehow connected to the signs that show up in graded commutative objects in topology?
Best Answer
There is a specific sort of situation I know about where that sign matters. Suppose you have $f:X \rightarrow Y$ and $g:Z \rightarrow W$ cofibrations (if the maps are not cofibrations, all the same things work - you just replace the quotient spaces by mapping cones). You extend both maps to their Dold-Puppe sequences, so you get the sequences
$X \rightarrow Y \rightarrow Y/X \rightarrow \Sigma X \rightarrow \Sigma Y \ldots$
and
$Z \rightarrow W \rightarrow W/Z \rightarrow \Sigma Z \rightarrow \Sigma W \ldots$
Now suppose you have maps $a: X \rightarrow W$ and $b: Y \rightarrow W/Z$ making the obvious square commute up to homotopy. You can then extend these to make a commutative ladder from the first Dold-Puppe sequence to the second. (Notice that the sequences are deliberately offset from each other by one spot.)
Using the usual parameters and the obvious choices of homotopies you will get a square involving $Y/X, \Sigma X, \Sigma Z, \Sigma W$. This square will commute if it includes the map $-\Sigma g: \Sigma Z \rightarrow \Sigma W$, but not generally with the map $\Sigma g$. (To check all this, I recommend doing the Dold-Puppe sequences with mapping cones rather than quotient spaces but keeping the homotopy equivalences with the quotient spaces in mind, which is the only way I know to calculate what the right maps should be.)
At this point, if you were feeling stubborn, you could replace the map in your ladder $\Sigma X \rightarrow \Sigma Z$ with $-1$ times that map, and that would allow you to have used $\Sigma g$ in the square I mention in the above paragraph, but that creates other issues; if you choose not to simply use suspensions of your original maps to go from one Dold-Puppe sequence to the other then you run into problem when you are mapping between Dold-Puppe sequences without the shift of this example.
I hope this helps unravel Greg's answer (which is correct - you need the sign to get good mapping properties).
Of course one sees exactly the same phenomenon in the category of chain complexes of abelian groups (where homotopy is chain homotopy) and other such categories. I agree with Theo and Mark that one thinks about the suspension as "odd" (in the sense of parity not the sense of peculiar).
The published paper that Mark refers to that has an error of exactly this sort (which is unfortunately fundamental to the paper) is by Lin Jinkun in Topology v. 29, no. 4, pp. 389-407. I read this paper in preprint form in 1988 and missed this error, but discovered it in 1992 when reading another paper by the same author with the same error. In the Topology paper the error is made in diagram 4.4 on the right hand square (proof of Lemma 4.3).