One version of the Freudenthal suspension theorem is the following:
Suppose a CW complex $X$ is a union of two subcomplexes $A,B$ with $A\cap B\neq\emptyset$ connected and nonempty. If $(A,A\cap B)$ is $m$-connected and $(B,A\cap B)$ is $n$-connected, then $$\pi_k(A,A\cap B)\cong \pi_k(X,B)$$ for $k<m+n$. Further, $\pi_{m+n}(A,C)$ surjects onto $\pi_{m+n}(X,B)$.
One can find proofs of this in Hatcher (theorem $4.23$) and Dieck (proposition $6.4.1$). My goal is to find some intuition for why this is true.
The corresponding statement in homology is excision, and the proof is fairly intuitive (although tedious). To show $$H_n(A,A\cap B)\cong H_n(A\cup B,B),$$ one can pretty easily hand-wave that, after subdivision, simplices in $A$ but not $A\cap B$ are the same as simplices in $A\cup B$ but not $B$. Hatcher's proof of Freudenthal suspension is vaguely reminiscent of this subdivision argument as well. By analogy, I have some intuition for why we have $\pi_k(A,A\cap B)\cong \pi_k(X,B)$. My main (soft) questions are the following:
- What is it about homotopy groups that makes excision fail in the first place? There are examples here, but I still can't really visualize them.
- What is the significance of $m+n$? This is my biggest source of confusion, because I really can't come up with any hand-waving argument that justifies the appearance of $m+n$.
The best hand-waving I have come up with is fairly pathetic: take a map $(D^{m+n},\partial D^{m+n})\to (X,B)$, and apply some simplicial structure so that the map is locally linear. Since $(A,A\cap B)$ is $m$-connected, you need to use $m+1$ of the dimensions to get into $A\setminus A\cap B$ (whatever that means). But then there are $n-1$ dimensions left, so they must map into $A\cap B$.
I don't even expect that to make sense to other people, and the dimensions are still off by one anyway. Your thoughts are appreciated!