I see on the webpage of a high school math summer program, SuMac, that they will cover some algebraic topology in a period of several weeks. And they covered every aspect of this subject, including Homology, Homotopy, Cohomology and so on.

I am a bit curious about how this is actually done in practice. After all, almost all algebraic topology texts assume quite a bit of knowledge about topologic and metric spaces. There are a lot of theorems and proofs to do even before starting algebraic topology.

Does anyone know why they are soooo efficient in teaching such difficult contents?

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    $\begingroup$ It would help to give some specific reference to your claims of what is said, especially since what you said does not seem to be consistent (e.g. "they will cover some algebraic topology" followed by "they covered every aspect of this subject"). Indeed, the descriptions here and here are VERY different from what you've described. Regarding how algebraic topology can be introduced at this level, see Algebraic topology in high school. $\endgroup$ – Dave L Renfro Jan 5 '20 at 16:00
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    $\begingroup$ I am a bit curious about how this is actually done in practice. --- A rather well known book (at one time, at least; probably mostly during the 1960s and 1970s) for introducing certain ideas of algebraic topology to very strong high school students is Chinn/Steenrod's First Concepts of Topology (freely available here). My best friend in high school covered this book in a 2 or 3 week selective summer program he attended in 1976. (I didn't apply, as I was taking classes at a nearby university.) $\endgroup$ – Dave L Renfro Jan 5 '20 at 16:26
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    $\begingroup$ I don't have first-hand knowledge of that program... but, with exceptional kids (who probably already know some abstract algebra, point-set topology, etc!), it would be easily possible to explain (as opposed to prove-everything-about) (for example) simplicial topology in a couple days. The idea of DeRham cohomology would also already be in the air, from multi-variable calculus. But, no, the "standard" Long March is surely not undertaken. (And I, for one, do not see why we do that to any students.) $\endgroup$ – paul garrett Jan 5 '20 at 19:48
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    $\begingroup$ @DaveLRenfro "Every aspect" doesn't contradict "some", because not everything in every part is covered. $\endgroup$ – Ma Joad Jan 5 '20 at 22:24
  • $\begingroup$ @DaveLRenfro a good book to mention, officially available at AMS/MAA press bookstore.ams.org/nml-18 $\endgroup$ – kcrisman Jan 10 '20 at 19:04

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