# Problem understanding May's proof of van Kampen

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:

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.

• 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. Oct 24, 2023 at 20:42
• @VincentBoelens I think I understand it now. Thanks tons for recommending!
– Atom
Oct 24, 2023 at 21:02
• @Atom Then you should write an answer to your own question. Oct 24, 2023 at 21:40
• @PaulFrost Done!
– Atom
Oct 26, 2023 at 18:43
• @Atom Nice answer! Oct 26, 2023 at 21:51

One should look at the following sequence of paths $$(0, 0)\to (1, 1)$$ in $$I\times I$$:
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 $$c_x\ast H_1$$.