Let me add some comment about the last theorem he stated; Every subgroup of a free group is a free group. He stated very roughly so maybe someone (including me) didn't understand. The following is stated in Hatcher textbook. We actually need some lemma maybe he omitted. ' Any covering space of a graph is a graph'. So eventually, by this lemma we can prove the theorem precisely: Given free group F, choose a graph that is isomorphic to F (e.g. choose wedge sum of S^1's). We know that for any subgroup G of F, for properly chosen base point, we get a covering space \tilda{X} and a covering map p such that p*(pi(\tilda{X})) = G. By injectivity of p_*, we get pi(\tilda{X}) isomorphic to G. Since \tilda{X} is a graph by lemma and the fundamental group of a graph is free so the statement follows.
This class follows Allen Hatcher's book Algebraic Topology very closely. It's available for free online. Many of the proofs the professor provides are taken directly from the book. This lecture is on section 1.3 from the textbook.
Thank you so much, we need more lectures of you please, you are so precise in your explanations.
Let me add some comment about the last theorem he stated; Every subgroup of a free group is a free group. He stated very roughly so maybe someone (including me) didn't understand. The following is stated in Hatcher textbook. We actually need some lemma maybe he omitted. ' Any covering space of a graph is a graph'. So eventually, by this lemma we can prove the theorem precisely: Given free group F, choose a graph that is isomorphic to F (e.g. choose wedge sum of S^1's). We know that for any subgroup G of F, for properly chosen base point, we get a covering space \tilda{X} and a covering map p such that p*(pi(\tilda{X})) = G. By injectivity of p_*, we get pi(\tilda{X}) isomorphic to G. Since \tilda{X} is a graph by lemma and the fundamental group of a graph is free so the statement follows.
It’s rather elegant how this basic theorem about free groups falls out naturally from covering spaces of graphs.
There is a book by Adrien Douady, Algebra and Galois theories that explores in detail Grothendieck's Galois theory.
Can a writing technology be found that makes the writing readable?
That would be chat GPT sir
This class follows Allen Hatcher's book Algebraic Topology very closely. It's available for free online. Many of the proofs the professor provides are taken directly from the book. This lecture is on section 1.3 from the textbook.