Three comments on Friday's posting regarding an exercise in Hatcher's text..........DMD ____________________________________________________________ Subject: [Fwd: Re: Questions about Algebraic Topology] From: Peter Landweber Date: Fri, 10 Dec 2004 17:35:31 -0500 -------- Original Message -------- Subject: Re: Questions about Algebraic Topology Date: Wed, 08 Dec 2004 13:25:07 -0500 From: Peter Landweber Organization: Rutgers University To: Lu Hwa Ooi References: <20041206222121.30030.qmail@web42205.mail.yahoo.com> Dear Lu Hwa, You have picked a good book to study algebraic topology. If you get really stuck, you can even e-mail the author. Although I have examined many of Hatcher's problems, I omitted to look at the exercises for section 2.B. Anyhow, my advice is to inspect closely the proof of Prop 2B.1, a somewhat intricate use of the Mayer-Vietoris sequence. Then practice on Exercise 1, and finally return to Exercise 2. Sincerely, Peter Landweber Lu Hwa Ooi wrote: >> >> Dear Professor Landweber, >> >> 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's my question: >> >> "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 >> reduced homology)" >> >> This is the problem 2 on page 176 in Hatcher. I am able to compute >> h_*(X) but I have no idea how to compute h_*(S^n \ X). >> >> Here is what I think: Let T be the maximal tree of X. Suppose X\T >> consists of n edges, >> then pi_1(X) is free on n generators, which means that h_1(X)=Z^n. >> Moreover, we can deduce that h_i(X)=0 for all i not equal to 1. But I >> have no idea of how to deduce h_i(S^n\ X)? Could you give me some >> suggestions? >> >> This may take your time. I'd appreciate if you could help me. >> >> Best Regards, >> Lu Hwa >> _______________________________________________________________ Subject: Re: question abt homology of complement From: Allen Hatcher Date: Sat, 11 Dec 2004 07:26:10 -0500 Since the question from the student in Malaysia involves an exercise from my book, perhaps it would be simplest for me to answer it. Rather than clutter up this discussion list with the solution, which won't be of interest to many people, I've made a little webpage with a sketch of the solution (in html): http://www.math.cornell.edu/~hatcher/AT/ATexercise2B.2.html Incidentally, this student had already sent me an email asking about this question and another one. Based on his comments and the fact that this exercise is well along in the book (on page 176), I replied with a couple brief hints about the solution, expecting this to be sufficient for him to solve it, but perhaps my hints were too cryptic. I certainly did not intend for the question to come to this discussion list. The webpage for the book mentions that I will try to answer queries, time permitting, about exercises in the book since solutions are not posted. I've gotten quite a few such queries from all sorts of interesting faraway places, and answering them hasn't been too great a burden so far, although it sometimes takes me a couple days. If I knew how to do it, it might be worthwhile to set up an interactive webpage for students to post and answer questions themselves. Allen Hatcher > 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 > _________________________________________________________________ Subject: Re: question abt homology of complement From: "Adrian P Dobson" Date: Mon, 13 Dec 2004 11:19:01 -0000 Hi, I'm a postgrad student at Manchester. Three of the people in my office have received emails very similar to the one just posted on your list. The only variation seems to be the question (usually from Hatcher), the name of the sender, and the country. I've copied the message I received below. It seems a little odd that (apparently) the same person is seeking help under several names. I am aware that certain universities use Hatcher as a text for graduate students, and am feeling a little suspicious. If this is part of someone's coursework, I'm not sure that I should be helping too much. Adrian ------- Forwarded message follows ------- Date sent: Sun, 5 Dec 2004 15:40:36 -0800 (PST) From: Herbert Subject: Nullhomotopic To: adobson@maths.man.ac.uk Dear Professor Dobson, I have a problem. Is the continuous map S^2 ---->S^1 nullhomotopic? To tell you the truth, I am a Singaporean and learn Algebraic Topology myself. It is difficult to find someone to discuss Algebraic Topology in Singapore so I searched the webpage of your university and got your email address. I hope you don't mind helping someone far away. Thanks, Herbert