Talk:Poincaré duality
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I have a problem with the line, ``kth homology group H^k(M) to the (n−k)-th cohomology group H_{n − k}(M).`` It is my understanding that homology groups are represented with a subscript, and cohomology with a superscript; reverse from what was written. I have changed this. If I am in error, please feel free to revert the edit and make a comment on the discussion page. 00:08 CST, 29 December 2004.
[edit] New To Advanced Math
Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as Poincaré duality, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006
[edit] Poincare's approach v.s. Modern
I really enjoyed this entry, thanks a lot! I think that perhaps it would be a small improvement to it if the following is taken into consideration:
You started by given a historical introduction, with Poincare's understanding of duality as concerned about Betti numbers. (I really enjoyed learning this). But them you stated the modern perpective, which is close to the first attempt, but it does not imply the "duality" between the Betti numbers right away. I think that it would be nice to reconcile the modern paragraph with the historical one by stating that by using the universal coeficients theorem the modern approach implies Poincare's approach.
