Hatcher in Chapter 4 gives the example: $\Sigma(S^m \times S^n) \simeq S^{m+1} \vee S^{n+1} \vee S^{m+n+1} $ relating the suspension and the wedge product.
for example $S^1 \vee S^1 $ looks like a figure eight.
$\Sigma(S^1 \times S^1) \simeq S^2 \vee S^2 \vee S^3 $
There's a couple of questions here. That wedge product is associative (in fact these two spaces are homeomorphic): $$ (S^2 \vee S^2) \vee S^3 \simeq S^2 \vee (S^2 \vee S^3) \tag{$\ast$}$$ The proof in the textbook also gives a homotopy equivalence:
$ (S^1 \ast S^1) \cup CS^1 \cup CS^1 \simeq \Sigma(X \times Y) $ where $\ast$ means join and $C$ means "cone".
It is the quotient space, $CX = (X \times I) /X \times \{ 0\})$.the join of two spaces is related to cone by $ X \ast Y \simeq (CX \times Y) \cup_{X \times Y} (CY \times X)$
I just wanted to picture what these shapes and maps look like. Here I'm just dealing with $m = n = 1$.
