Let $X$ be a finite CWcomplex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. The quotient complex $Y^n/X^n$ has the cell structure with one cell for each product of cells $e_1\times\cdots\times e_n$ where the $e_i$ are either equal to $x_i$ or $y$ and at least one $e_i$ is equal to $y$. I want to say that this amounts to a wedge of products of spheres each product of spheres depending on how many copies of $y$ there is in that cell. Is this correct?
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Suppose Y is obtained by attaching a zerocell, so $Y = X \cup \{\ast\}$. Then $Y^2$ is $$(X \times X) \cup (X \times \{\ast\}) \cup (\{\ast\} \times X) \cup (\{\ast\} \times \{\ast\})$$ and so $Y^2/X^2$ is homeomorphic to $$ \{\ast\} \cup X \cup X \cup \{\ast\}. $$ This can be arbitrarily complicated depending on X. ADDENDUM: By request, a connected example is the inclusion $\mathbb{CP}^1 \subset \mathbb{CP}^2$. The quotient $Y^2/X^2$, in this case, has a nonzero cohomology operation $Sq^2$ from H^{6} to H^{8} with ℤ/2coefficients, and there is no wedge of products of spheres that can have this cohomology. (You should work out the details.) 

