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I am reading the proof of van Kampen's theorem that May gives in his book, particularly its fundamental groupoid version. I am able to follow through all of it, except the last part when he says:

We see that the relation $[f] = [g]$ in $\Pi(X)$ is a consequence of a finite number of relations, each of which holds in one of the $\Pi(U)$. Therefore $\tilde\eta([f]) = \tilde\eta([g])$.

(Let me know if I should post the whole proof for reference.)

Can someone help by enunciating what exactly the "relations" are that hold in respective $\Pi(U)$'s?

I do understand that in each subsquare, the bottom edge will be homotopic to the top edge in some $U$ and thus their $\eta$'s will be the same.
The trouble I am having is in moving across the squares in a given column and deducing that $\eta_U([f_i]) = \eta_V([g_i])$ where $U$ is an open set from $\mathscr O$ that contains the image of the bottom edge of the bottom-most subsquare of the column, and $V$ is one that contains the image of the top edge of the top-most subsqaure of the column. The following image shows how I am thinking:

enter image description here


Incidentally, the same question was asked before. But the accepted answer there doesn't seem to be correct, saying that $I\times I$ is being divided into vertical strips, and not subsquares.

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    $\begingroup$ The book by tom Dieck has more details. Essentially, what you do is consider homotopies that only differ on one of the subsquares and then do induction. $\endgroup$ Oct 24, 2023 at 20:42
  • $\begingroup$ @VincentBoelens I think I understand it now. Thanks tons for recommending! $\endgroup$
    – Atom
    Oct 24, 2023 at 21:02
  • $\begingroup$ @Atom Then you should write an answer to your own question. $\endgroup$
    – Paul Frost
    Oct 24, 2023 at 21:40
  • $\begingroup$ @PaulFrost Done! $\endgroup$
    – Atom
    Oct 26, 2023 at 18:43
  • $\begingroup$ @Atom Nice answer! $\endgroup$
    – Paul Frost
    Oct 26, 2023 at 21:51

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I had a look at Dieck's book that @Vincent suggested, and it cleared things for me.

One should look at the following sequence of paths $(0, 0)\to (1, 1)$ in $I\times I$:

enter image description here

These in turn give a sequence of paths $x\to y$ in the space $X$ (when the homotopy $H$ is applied). The upshot is this: Any two consecutive paths in $I\times I$ differ by a subsquare whose $H$-image is contained in some $U$, and thus their $H$-images will be path homotopic in that $U$, rendering their $\eta$-expressions the same. Now, just note that the first path in the sequence is just $H_0\ast c_y$ and the last path in the sequence is $c_x\ast H_1$.

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