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I am coming from this post and i am fine with everything that happens there. I just want a few more details on the calculation of the effect of $d_2$. Following the way i learnt it we have to calculate the degree of the map $S^1\overset{\phi_{\delta \alpha}}{\rightarrow} S^1\vee S^1 \overset{c_{\alpha'}}{\rightarrow} S^1$ where $\phi_{\delta \alpha}$ is the attaching map given by $aba^{-1}b^{-1}$ and $c_{\alpha'}$ collapses the respective circles (so $c_{\alpha'} $ varies. Now im stuck at making explicit what this actually means. Intuitively it is quite clear what is going on but its hard for me to make it rigorous. Any help is apprechiated!

Adronic
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  • Degree of $d_2$? In a homological chain complex, the degree of the boundary map is always $-1$. You didn't mean to say 'degree', right? – feynhat Feb 27 '23 at 16:53
  • In any case, you need to compute the image and kernel of $d_1$ and $d_2$. Which is what the linked post does.Can you elaborate on the 'way you have learnt it'? – feynhat Feb 27 '23 at 16:56
  • Yeah i did not mean to say that, thank you. Well calculating the kernel and image is easy when you have the map. i think the difficult part is always to calculate the effect of the boundary maps explixitely. I have learned it as follows: You fix a basis of Generators in the cellular chain complexes of the n-skeleta. Then you want to know the effect that your boundary map has with respect to those basis and this is essentially the same as calcuating the degree of the map in my question. I want to get really explicit! – Adronic Feb 28 '23 at 09:19
  • What is a way you know to compute the degree of a map $S^1 \to S^1$? – ronno Feb 28 '23 at 12:07
  • I basically know two ways. One is that you track it back to some degree you already know. The other one is to use the local degree formula. – Adronic Feb 28 '23 at 22:38
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    The local degrees at a point in the part of the boundary corresponding to $a$ will be the opposite of that at a point in the part corresponding to $a^{-1}$ because the maps near those points differ by a reflection of the square, which is a degree $-1$ map of the boundary $S^1$. – ronno Mar 01 '23 at 12:14
  • Well thats exactly what i am looking for and i did realize that as well, but how does one formalize that thought? Which point precisely do you take? just any random point? It would be really nice if you could provide an answer which is formal. – Adronic Mar 01 '23 at 13:03

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