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The excercise wants us to show that for CW complexes $X$ and $Y$ with $0$-cells $x_0$ and $y_0$, we have a homeomorphism between the two spaces $$X*Y/(X*\{y_0\} \cup \{x_0\}*Y) \cong S(X \wedge Y)/S(\{x_0\} \wedge \{y_0\}).$$ I was reading the proof in this post here as well as the solution found here. Both began with looking at the subspace $$I \times (X \times \{y_0\} \cup \{x_0\} \times Y) = (I \times X \times \{y_0\}) \cup (I \times \{x_0\} \times Y) \subset I \times X \times Y$$ and deduced that it gets identified to the subspace $X*\{y_0\} \cup \{x_0\}*Y$ in $X*Y$ which is completely fine. Then both posts claims that we automatically get a homeomorphism \begin{equation}\label{eq1}\frac{I \times X \times Y}{I \times (X \times \{y_0\} \cup \{x_0\} \times Y)} \cong \frac{X*Y}{X*\{y_0\} \cup \{x_0\}*Y}.\end{equation} My question is, why do we have this? I am assuming the quotients just mean the subspaces $I \times (X \times \{y_0\} \cup \{x_0\} \times Y)$ and $X*\{y_0\} \cup \{x_0\}*Y$ are being collapsed to a point in the respective spaces.

However, what about points of the form $(0,x_1,y_1) \in I \times X \times Y$ for $x_1 \neq x_0$ and $y_1 \neq y_1$? When constructing $X*Y$, such point is identified with $(0, x_1, y_0)$ since $\{0\} \times X \times Y$ gets collapsed to $\{0\} \times X$ in $X*Y$. However, in the above homemorphism, since such a point $(0,x_1,y_1) \notin I \times (X \times \{y_0\} \cup \{x_0\} \times Y)$ i.e. it is not identified with anything on the LHS after taking the quotient whereas, it is identified in the RHS when constructing $X*Y$. So how can the two quotient spaces be homeomorphic?

KSAKY
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    Solution manual? Something some student wrote up and made it sound like Hatcher deemed it correct and valid? – Ted Shifrin Feb 22 '24 at 21:44
  • @TedShifrin I have corrected the link description. So are you claiming that their 'proofs' aren't correct? – KSAKY Feb 22 '24 at 22:13
  • This is the first time I've seen this for that book, and I haven't read the solution. I do know that I've come across things on the web claiming to be solutions manuals for textbooks I have written, and they are totally unauthorized and, from the few items I've checked, far from what would be in a true solutions manual. These are some individuals or companies trying to bilk the public. – Ted Shifrin Feb 22 '24 at 22:47
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    Indeed, that "author" has posted "solutions" to various problems in various textbooks. Google research shows that he earned a doctorate in mathematics, with a concentration in cryptography. But to post his smattering of textbook solutions as solutions manuals is pompous and, IMHO, not to be trusted. – Ted Shifrin Feb 22 '24 at 22:55
  • Anyhow, back to your question. It seems that the basic quotient construction by which we get from $I\times X\times Y$ to $XY$ has been lost. It is definitely not* identifying the quotienting subspace to a point, as you suggest. – Ted Shifrin Feb 22 '24 at 23:51
  • @TedShifrin Thanks for this. So are you saying that the original identification done when constructing $X*Y$ from $I \times X \times Y$ should still be done when forming the quotient space $\frac{I \times X \times Y}{\text{that subspace}}$? – KSAKY Feb 23 '24 at 00:12
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    Yes, I certainly would say so. Try writing the original equivalence relation as $\sim$ and keep track of it explicitly. – Ted Shifrin Feb 23 '24 at 01:46

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