What are examples of cellular decomposition?

quasar987

Say X is a CW-complex. Then for any n, the n-skeleton X^n of X is obtained from the (n-1)-skeleton X^(n-1) by gluing some n-cells on X^(n-1) along their boundary.

From what I read, it seems that the way to obtain X^n from X^(n-1) in this way is not unique.

Is this non-uniqueness 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^(n-1) by using a different number of n-cells?

wofsy

cell decompositions are not unique.

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.

quasar987

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 CW-complex X has dimension n (meaning the maximum dimension of cells is n), then it is obtained from a (sub-)CW-complex X^(n-1) of dimension n-1 by attaching n cells to it. Is it possible to get X from X^(n-1) in two ways that involve a different amount of n-cells?

I'm guessing no but I don't see how to prove this.

Oh, I just noticed that the open n-cells in X are precisely the connected components of X\X^(n-1) so building X from X^(n-1) with a different numbers of n-cells is impossible!