Register to reply

What are examples of cellular decomposition?

by quasar987
Tags: cellular, decomposition, examples
Share this thread:
quasar987
#1
Feb12-09, 04:42 PM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,771
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?
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
wofsy
#2
Feb14-09, 11:00 AM
P: 707
Quote Quote by quasar987 View Post
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?
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
#3
Feb15-09, 01:22 PM
Sci Advisor
HW Helper
PF Gold
quasar987's Avatar
P: 4,771
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!


Register to reply

Related Discussions
Cellular Membranes Biology 7
Cellular Atom Set Theory, Logic, Probability, Statistics 1
Cellular memory General Discussion 2
Cellular Respiration Introductory Physics Homework 1