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Use the Mayer-Vietoris sequence to show there are isomorphisms $\tilde H_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde H_n(Y)$ if the basepoints of $X$ and $Y$ that are identified in $X \vee Y$ are deformation retracts of neighborhoods $U \subset X$ and $V \subset Y$.

So for this one, I am thinking of writing $\tilde H_n(X \vee Y)$ in the form of $\tilde H_n((X \cup U) \cup (Y \cup V))$ in order to satisfy the Mayer-Vietoris sequence condition, that $X \vee Y$ is the union of interior of $X \cup U$ and $Y \cup V$. But I am not sure how to justify it?

1LiterTears
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  • Since $U \subset X$, we have $X \cup U = X$. What you want is $X \cup V$ and $Y \cup U$. Use the definition of the quotient topology to justify that both are open. – Ayman Hourieh Nov 17 '13 at 10:50

1 Answers1

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This is almost right except you've taken an essentially redundant union (of $X$ and $U$, and $Y$ and $V$). What you want to use as your two spaces is $W=X\cup V$ and $Z=Y\cup U$ which have intersection equal to $U\cap V$ which clearly deformation retracts first to $U$ (or $V$), and then to the wedge point. You can also see that $W$ is homotopy equivalent to $X$, and $Z$ is homotopy equivalent to $Y$.

Dan Rust
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  • Oh exactly! That's why I got stuck! Thank you so much Daniel! – 1LiterTears Nov 17 '13 at 23:40
  • No problem! Glad to help. – Dan Rust Nov 17 '13 at 23:44
  • Just to make it clear: So the wedge point was on the boundary of $X$ and $Y$? I am not sure that why we can just chose $X$ and $Y$, but we need $W$ and $Z$. – 1LiterTears Nov 18 '13 at 00:52
  • Consider $X=Y=([0,1],1)$. You can't choose $X$ and $Y$ to be the spaces used in the application of Mayer-Vietoris because the interior of $X$ and the interior of $Y$ (seen as subset of their wedge sum) do not cover the wedge $X\vee Y=[0,1]\times{0}\sqcup [0,1]\times{1}/((1,0)\sim(1,1))$ – Dan Rust Nov 18 '13 at 01:02
  • I see. That is clear now. Thank you so much Daniel! – 1LiterTears Nov 18 '13 at 01:11
  • Although this was posted ages ago, do you mind explaining the intuition that might lead one to choosing these sets? – Anon Feb 14 '23 at 14:05
  • We've these types of questions, sometimes the easiest approach is to just look at the information provided and see what you can do with them. In this case, there are very few choices for what to take as your two spaces in terms of unions and intersections of the ones appearing in the question, and even fewer that give something useful when you take their intersection (in this case it was clear that $U \cap V$ is contractible, so we want that to appear somewhere). Of course, if you draw the picture of the above setup, it also makes it easier to see what spaces one might want to take. – Dan Rust Feb 14 '23 at 22:16