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sniffnoyAck. So algebraic topology homework, due today. I'm starting it very late, and due to having missed class, I need to read about the stuff in the book. Computing homology groups of CW complexes. I get how to do it for 2-dimensional, but I don't really have a handle on higher-dimensional. 3rd problem, what do you know, compute the homology groups of this 3-dimensional CW complex.
OK. It has 1 0-cell, 1 1-cell, 1 2-cell, 2 3-cells. So 0→Z²→Z→Z→Z→0 via d3, d2, d1. 1 0-cell so d1 is 0. d2 I know well how to compute, as it happens it's n→2n. Now d3... this will be more of a problem...
OK, in the book, it seems to do this by looking at local degrees. So, I do this... except, uh, how do I actually compute these local degrees? Well, this is a local homeomorphism, right? So they must be ±1... but what do they total to? Well, I thought they ought to total to ±2, so I write down a quick explanation of why they have the same sign, only, is this right? I have no idea. In fact, it looks like I just may have it backwards and they have opposite signs... I don't have much time so I ignore this. This means our homomorphism is (n,m)→2(n+m).
So! H3(X) is ker d3≅Z, H2(X) is ker d2/Im d3, which is... wait a minute here...
D'oh! Clearly I've messed up. By my calculations, Im d3 (2Z) isn't contained in ker d2 (0)! What? Class is beginning so I leave it at that, with no answer.
Only during class do I realize that all this calculation of what d3 is was entirely unnecessary. The condition that Im d3⊆ker d2 tells us right there that d3 is 0! So we get homology groups Z², 0, Z2, Z.
GAH.
Though I have to wonder how we were supposed to do a later problem, another computation of homology groups of a 3-dimensional CW complex, where d2 was not 0, seeing as apparently we hadn't done local degrees in class yet.
-Harry
OK. It has 1 0-cell, 1 1-cell, 1 2-cell, 2 3-cells. So 0→Z²→Z→Z→Z→0 via d3, d2, d1. 1 0-cell so d1 is 0. d2 I know well how to compute, as it happens it's n→2n. Now d3... this will be more of a problem...
OK, in the book, it seems to do this by looking at local degrees. So, I do this... except, uh, how do I actually compute these local degrees? Well, this is a local homeomorphism, right? So they must be ±1... but what do they total to? Well, I thought they ought to total to ±2, so I write down a quick explanation of why they have the same sign, only, is this right? I have no idea. In fact, it looks like I just may have it backwards and they have opposite signs... I don't have much time so I ignore this. This means our homomorphism is (n,m)→2(n+m).
So! H3(X) is ker d3≅Z, H2(X) is ker d2/Im d3, which is... wait a minute here...
D'oh! Clearly I've messed up. By my calculations, Im d3 (2Z) isn't contained in ker d2 (0)! What? Class is beginning so I leave it at that, with no answer.
Only during class do I realize that all this calculation of what d3 is was entirely unnecessary. The condition that Im d3⊆ker d2 tells us right there that d3 is 0! So we get homology groups Z², 0, Z2, Z.
GAH.
Though I have to wonder how we were supposed to do a later problem, another computation of homology groups of a 3-dimensional CW complex, where d2 was not 0, seeing as apparently we hadn't done local degrees in class yet.
-Harry
no subject
Date: 2007-05-30 12:34 am (UTC)