What are examples of cellular decomposition?

In summary, for any CW-complex X of dimension n, the n-skeleton X^n is obtained from the (n-1)-skeleton X^(n-1) by attaching n cells to it. This process is not unique, but it is impossible to obtain X from X^(n-1) using a different number of n-cells due to the definition of open n-cells in X.
  • #1
quasar987
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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?
 
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  • #2
quasar987 said:
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.
 
  • #3
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!
 

Related to What are examples of cellular decomposition?

1. What is cellular decomposition?

Cellular decomposition is the process by which cells break down and are recycled by the body. This occurs naturally during cellular death or during normal bodily functions such as digestion.

2. What are some examples of cellular decomposition?

Examples of cellular decomposition include the breakdown of red blood cells in the spleen, the breakdown of muscle cells during exercise, and the breakdown of food particles in the digestive tract.

3. How does cellular decomposition affect the body?

Cellular decomposition is an important process for maintaining the overall health and function of the body. It allows for the recycling of important nutrients and the removal of waste products.

4. Is cellular decomposition the same as cell death?

Cellular decomposition is a natural process that occurs during cell death, but it is not the same thing. Cell death can occur due to injury or disease, while cellular decomposition is a gradual and controlled breakdown of cells.

5. Can cellular decomposition be prevented or slowed down?

While cellular decomposition is a natural and necessary process, there are certain factors that can speed it up or slow it down. A healthy diet and lifestyle, as well as proper medical care, can help slow down the process of cellular decomposition.

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