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What are examples of cellular decomposition? 
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#1
Feb1209, 04:42 PM

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PF Gold
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Say X is a CWcomplex. Then for any n, the nskeleton X^n of X is obtained from the (n1)skeleton X^(n1) by gluing some ncells on X^(n1) along their boundary.
From what I read, it seems that the way to obtain X^n from X^(n1) in this way is not unique. Is this nonuniqueness superfluous (in the sense that only the way in which the cells are attached can differ), or are there really examples where one can obtain X^n from X^(n1) by using a different number of ncells? 


#2
Feb1409, 11:00 AM

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for instance, the 2 sphere is a 2 disk whose boundary is attached to a point. it is also a circle attached to a point then two 2 disks attached to the circle along their boundaries. 


#3
Feb1509, 01:22 PM

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Hello wofsy and thanks for the reply.
But I don't think the example that you give answers my question. Let me rephrase it. If a CWcomplex X has dimension n (meaning the maximum dimension of cells is n), then it is obtained from a (sub)CWcomplex X^(n1) of dimension n1 by attaching n cells to it. Is it possible to get X from X^(n1) in two ways that involve a different amount of ncells? I'm guessing no but I don't see how to prove this. Oh, I just noticed that the open ncells in X are precisely the connected components of X\X^(n1) so building X from X^(n1) with a different numbers of ncells is impossible! 


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