I'm learning Van Kampen theorem but don't see any its applications . Can anybody give me some exercises ? Example , compute some fundamental groups .
How to use Van Kampen theorem to compute some fundamental groups?
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general-topology
algebraic-topology
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0From where do you learn about this theorem? It is very useful, and usually any book that teaches it also come with examples. For example - take the 2 dimensional sphere and decompose it to two disks where the intersection is a ring, to get that the fundamental group is trivial. – 2017-01-25
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0Try chapter 1.2 of https://www.math.cornell.edu/~hatcher/AT/AT.pdf. Lot's of exercises at the end. – 2017-01-25
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0I read in Munkres's book , but anyway thank you so much . – 2017-01-25
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0@Gankedbymom, You can see Massey's book "A Basic Course in Algebraic Topology" Chapter 4 is devoted to applications of Van Kampen – 2017-01-25
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0If you want a van Kampen Theorem which computes the fundamental group of the circle $S^1$, a rather basic example in algebraic topology, then the only available topology text is at http://groupoids.org.uk/topgpds.html – 2017-01-25
1 Answers
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As already said in the comments, a good reference is the book of Allen Hatcher which contains plenty of exercises.
Originally I believe the Van-Kampen theorem was created for computing fundamental group of complements of algebraic planes curves but this is probably a bit technical.
The most simple (and probably one of the most useful) applications of Van Kampen is to compute the fundamental group of a wedge product. You can also draw a graph and compute its fundamental group.