8.06 Galois correspondence, 2

Below the video you will find accompanying notes.


(0.00) In this section, we will put together everything we have seen about covering spaces to get the Galois correspondence between covering spaces and subgroups of the fundamental group.

Covering spaces give subgroups of \(\pi_1(X,x)\)

(0.13) A covering space \(p\colon Y\to X\) together with a point \(y\in p^{-1}(x)\) gives a subgroup \(p_*\pi_1(Y,y)\subset \pi_1(X,x)\). We have also just seen that if there is a simpy-connected (universal) cover then any subgroup arises this way. This assignment \((Y,y)\mapsto p_*\pi_1(Y,y)\) is called the Galois correspondence between (based) covering spaces and subgroups of \(\pi_1(X,x)\). There is some more (functorial) structure, which allows us to read properties of the covering space off from properties of the subgroup, which we will now explain.

(1.18) We know that \(\beta p_*\pi_1(Y,y)\beta^{-1}=p_*\pi_1(Y,\sigma_\beta(y))\). This tells us conjugates of subgroups are represented by the same covering space but with a different basepoint. More precisely, if \(y,y'\in Y\) then a path between \(y\) and \(y'\) projects to get a loop \(\beta\) in \(X\) such that \(y'=\sigma_\beta(y)\) so a single covering space \(Y\) defines a conjugacy class of subgroups of \(\pi_1(X,x)\), and any subgroup in that conjugacy class arises as \(p_*\pi_1(Y,y)\) by picking \(y\) suitably.

(3.27) We know that there is a covering transformation \(F\colon Y_1\to Y_2\) with \(F(y_1)=y_2\) if and only if \[(p_1)_*\pi_1(Y_1,y_1)\subset (p_2)_*\pi_1(Y_2,y_2).\]

Let's draw a picture to represent this fact as follows. For each (based) covering space \((Y,y)\) of \(X\), we draw a dot labelled \((Y,y)\). For each covering transformation between covering spaces \(F\colon Y_1\to Y_2\), \(F(y_1)=y_2\), we draw an arrow from the dot \((Y_1,y_1)\) to the dot \((Y_2,y_2)\). Now if we replace each dot \((Y,y)\) by the subgroup \(p_*\pi_1(Y,y)\), we can replace each covering transformation by an inclusion. This means that the Galois correspondence is a functor from the category of based covering spaces (with covering transformations) to the category of subgroups (with inclusions). This is a lot like the Galois theory of field extensions, in which field extensions correspond to subgroups of the Galois group.

(6.20) Finally, we also saw that \(Deck(Y,p)=N_H/H\), where \(H=p_*\pi_1(Y,y)\) and \(N_H\) denotes the normaliser of \(H\subset\pi_1(X,x)\). This is again very reminiscent of the Galois theory of field extensions.


(7.30) Take \(X=S^1\) and \(x=1\in S^1\). We have \(\pi_1(X,x)=\mathbf{Z}\). The subgroups of \(\mathbf{Z}\) are: Some of these subgroups are nested, for example \(4\mathbf{Z}\subset 2\mathbf{Z}\), which means that the corresponding covering spaces (say \(Y_4\) and \(Y_2\)) are related by covering transformations \(Y_4\to Y_2\). Inside all of these subgroups we have the trivial group, so the covering space \(\mathbf{R}\) covers all of them.

(11.32) Take \(X=S^1\vee S^1\) and \(x\) the cross point. We have \(\pi_1(X,x)=\mathbf{Z}\star\mathbf{Z}\). (17.24) There is an inclusion \(\langle a\rangle\to norm(a)\). Under the Galois correspondence, this gives a covering transformation from the cover \(Y_1\) for \(\langle a\) to the cover \(Y_2\) for \(norm(a)\). This covering transformation is illustrated in the video.

Of course, there are many other subgroups of \(\mathbf{Z}\star\mathbf{Z}\), and, correspondingly, many other covering spaces.


Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.