8.06 Galois correspondence for covering spaces 2. Summary and examples

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We bring together various things we have proved to state the Galois correspondence for covering spaces and illustrate this with some examples.

Jonathan Evans

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Thank you very much for this video and for the whole playlist, which I have enjoyed a lot!

I liked that you ended the series by providing examples. In particular, you addressed the point I raised in a comment to some earlier video about the existence of a universal cover.

The universal cover of the figure 8 is exactly the example I tried to describe in that comment. I had wrongly named it “free space” (basically, I was looking at it as a free tree construction, if that makes sense). It’s kind of funny because the figure 8 is what corresponds to the free group on two generators. While its universal cover corresponds to the trivial group (which could also be seen as a free group, I guess, but without generators).

Back to the main point. At the end of the video, you hinted at the fact that there are spaces that don’t admit a universal cover. I thought about several options (using variants of the topologist’s sine curve, one of the weirdest spaces I know). But so far I haven’t been able to find a good counter-example. I will do some research then.

Thanks again!

lfossati

This is a great course and is being very useful to me in the study of algebraic topology, I would love to see a video on the construction of the universal cover

laflaca

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