Is wedge sum for finite CW complexes cancellative in the homotopy category?

Solution 1:

Yes, it is possible, and so wedge sum for finite CW complexes does not satisfy homotopy cancellation. There is a description of some examples in the article "On principal $S^3$-bundles over spheres" by P. Hilton and J. Roitberg, although they mention they already appear in "On the Grothendieck group of compact polyhedra" by P. Hilton. Here is the idea of the construction:

Given a pointed map $\alpha \colon S^{m-1} \to S^n$ consider the mapping cone $C_{\alpha} = S^n \cup_{\alpha} D^m$. Assume that $\alpha$ is a suspension and the class $[\alpha] \in\pi_{m-1}(S^n)$ has prime order $p \neq 2,3$. Let $l$ be prime to $p$ and not congruent to $\pm 1$ mod $p$. Then, if $\beta \colon S^{m-1} \to S^n$ is a pointed map representing the class $ l[\alpha]$, then

$$ C_{\alpha} \vee S^m \simeq C_{\beta} \vee S^m $$

$$ C_{\alpha} \vee S^n \simeq C_{\beta} \vee S^n$$

but $C_{\alpha}$ is not homotopy equivalent to $C_{\beta}$.

We can not use this construction for $S^2$, for there are no elements in $\pi_k(S^2)$ nor in $\pi_1(S^n)$ that are suspensions and have prime order.

On the other hand, the article "On the Grothendieck group of compact polyhedra" also contains the following result. If $X$, $Y$ and $A$ are compact connected polyhedra that are suspensions and such that $X \vee A \simeq Y \vee A$, then the homotopy groups of $X$ and $Y$ are isomorphic.

The dual question is also approached, whether Cartesian product satisfies homotopy cancellation and the answer is also no, examples are constructed in the article "On principal $S^3$-bundles over spheres" using a construction which is a dual in some sense to the one given above.

There are many articles which refer to these two and which contain similar results.