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I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or proof would be much appreciated, but it is possible my guess is off so any direction towards a counterexample would of course be useful as well.

From now we assume all CW-complexes to have a single $0$-cell.

Given a CW-pair $(Y,X)$, we form $X’$ by gluing some amount of new cells to $X$ (where these new cells are arbitrary and not cells of $Y$). We denote by $Y'$ the result of gluing in these same cells to $Y$ by gluing them into the subcomplex $X\subset Y$. We ensure in doing this that the new cells do not intersect $Y$ outside of this gluing into $X\subset Y$. The question is do we have that $Y/X$ is homeomorphic to $Y'/X'$?

Some notes so far: the cells of $Y/X$ can be identified with those of $Y-X$ and a $0$-cell, and likewise for $Y'/X'$, so in particular these quotient spaces have the same cells (in the sense that there is a natural identification between the cells of each space). Hence it suffices to check that the attaching maps are the same through this identification. I think I can prove the result for when $X$ and $Y$ are 2-complexes, but I am interested as to whether it is true in some higher dimensions as well.

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  • $\begingroup$ I don’t understand the question. Are the cells you are gluing to X already part of Y? What does “gluing in these same cells to $X\subset Y$ such that they do not intersect $Y$ outside of $X$” mean, precisely? [For those voting to close, could we please give the OP a chance to clarify what he’s asking?] $\endgroup$
    – Dan Ramras
    Commented Apr 11 at 11:21
  • $\begingroup$ Apologies for the confusion, will edit the question to make it more clear. The idea is we take $Y$ and then glue in the cells we glued to $X$ (when we made X’) to the subcomplex $X\subset Y$. Whilst doing this we ensure that the new cells glued onto $Y$ only intersect $Y$ at the points of $X\subset Y$ where we define the original gluing. $\endgroup$
    – William Thomas
    Commented Apr 11 at 14:15
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    $\begingroup$ I recommend the first appendix of Hatcher's Algebraic Topology book for this kind of general topology issues. $\endgroup$
    – Fernando Muro
    Commented Apr 11 at 14:21
  • $\begingroup$ @DanRamras to answer you first question the cells we attach to $X$ to get $X’$ are arbitrary and not related to $Y$ in any way, although I appreciate the way I wrote it could have seemed to imply this. $\endgroup$
    – William Thomas
    Commented Apr 11 at 14:23
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    $\begingroup$ @WilliamThomas I think that Proposition A.2 in the aforementioned book by Hatcher should solve your problem. $\endgroup$
    – Fernando Muro
    Commented Apr 11 at 14:30

1 Answer 1

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If I understand the question correctly, you have a CW complex $Y'$ which is the union of two subcomplexes $Y$ and $X'$ whose intersection is the subcomplex $X$. We can first collapse $X$ to a point to get $Y'/X$ as the wedge sum $Y/X \vee X'/X$. Then when $X'/X$ is collapsed to a point one gets $Y/X$. This two-stage collapsing is equivalent to the forming the single collapse $Y'/X'$. So $Y'/X' = Y/X$.

To check that there is no problem with the point-set topology, one can compare the characteristic maps for $Y'/X'$ and $Y/X$, which are really the same maps.

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