Subject: a question for the list From: "Daniel Alayón Solarz" Date: Fri, 10 Dec 2004 18:31:35 -0200 (BRST) I have received the following question from Lu Hwa Ooi. The question, seems to me, is well-formulated. As Lu is a self-taught young mathematician, with acces to the internet, I think he is not merely looking for the solution for this particular question but also in getting advice about interesting references and how this problem can or not be of importance to the algebraic topology. I remember following some interesting discussion on the list precisely about what should the best references when introducing some important areas to young students and the importance of starting with interesting calculations right over. I would like to add that free-internet references (i.e Hatcher) would be prefered by Lu over the rest as long as it is possible. Moreover, one of my advisors; Carlos Duran, prepares himself to lecture on Homology (usually following Hatcher) for the next year new phd students. I am therefore specially interested on this subject too. I hope members of the list could help Lu and eventually me. Regards, Daniel Alayon-Solarz -------- Original Message -------- Subject: S^n From: Lu Hwa Ooi Date: Wed, December 8, 2004 8:41 am To: solarz@ime.unicamp.br Dear Dr Alayon-Solarz Let me introduce myself first. I am a Malaysian and learn Algebraic Topology myself. As it is difficult to find someone who is familiar with Algebraic Topology in Malaysia so I searched the webpage of your university and got your email address. I hope you don't mind helping someone far away. Here is my question(indeed it is from Hatcher) "Suppose X is a subset of S^n that is homeomorphic to a finite connected graph, then h_i(S^n \ X) is isomorphic to h_{n-i-1}(X) for all i. ( h_*denotes the i-th reduced homology group)." I have showned that if T is a maximal tree of X and X\T consists of k edges, then h_i(X)=Z^k if i=1 and 0 otherwise. However, finding the reduced homology of S^n\X is much difficult. Perhaps we can prove by induction on k and use Mayer-Vietoris sequence. But this doesn't suffice to prove the desired isomorphism. Is there any fact I have missed? Hopefully you can give some suggestions. Thanks, Lu Hwa -- Daniel Alayón Solarz PhD Student UNICAMP - IMECC Praça Sergio Buarque de Holanda, 651 Cidade Universitária - Barão Geraldo Caixa Postal: 6065 13083-859 Campinas, SP, Brasil