# Exhaustive lists of CW complex structures

The 1-sphere can certainly be built from a 0-cell and a 1-cell. Also from two 0-cells and two 1-cells. These are the canonical n-sphere structures as well.

Method 1: take an n-cell and a 0-cell.
Method 2: take 2 i-cells in each dimension with $0 \leq i \leq n$.

Of course there are infinitely many CW structures that describe any given $S^n$ with n finite. Countably infinite, I think, and this should be the case for the exhaustive list of any finite dimensional CW complex.

What I'm wondering is if this list can generally be described in closed-form.

For $S^1$, the only structures are, with $i>0$, i 0-cells and i 1-cells. Similarly for $D^1$, but with one less 1-cell.
Inductively, we can then describe an exhaustive list for the cell structure of $S^n$ by considering any number of $S^{n-1}$ (possibly with different cell structures) as 'latitudes' of $S^n$, which we can connect with $(n-1)$-cells and fill in with $n$-cells. Similarly for double attaching $n$-cells to copies of $D^n$. (We also need to include a 'degenerate' structures, like the $0$-cell/$n$-cell construction that isn't obtained inductively.)

So, although I don't want to write this out right now, all countably-many CW-complexes structures for a finite dimensional n-sphere can be enumerated by this process.

Is it generally possible to enumerate all the CW cell structures for an arbitrary finite-dimensional CW space? (up to the number of cells in each n-skeleton, ignore the attaching maps...)

Furthermore, what kind of information can be extracted from this list? There must be so much information contained in an exhaustive CW description of a space, even if we forget about the specifics of attaching for any particular cell structure. I really like the idea of describing topological structure with purely integer data.

• of course, this idea would be immediately invalidated if someone provided an example of homotopically distinct spaces X and Y with CW structure such that their exhaustive cell-structure lists were identical, meaning that attachment information just cannot be ignored. – Badam Baplan Dec 1 '14 at 2:24

## 1 Answer

You're not going to find an exhaustive list, because there are infinitely many possible CW-complexes homeomorphic to even $S^1$. (For any positive integer $n$, take as the 0-skeleton the $n$th roots of unity.)

A way to approach things is that a complex (either simplicial or CW) is a method of imposing a combinatorial structure on a topological space in such a way as to make extraction of homological information relatively straight forward. In the case of a simplicial complex, extracting the information is trivial, but the complex can be unwieldy (simplicial complexes behave poorly with respect to products, for example). CW complexes are meant to allow a minimal analogue, so trying to find complicated CW decompositions somewhat misses the point.

• I definitely see what you mean about missing the point. This is why none of the introductory Algebraic Topology texts I've come across address this issue. But I still find myself wondering if the possible CW-complex structures for some spaces (I chose $S^n$ here for simplicity) couldn't be expressed in close form, even if this list is infinite. Also, when is this list countably infinite? When is it uncountably so? My thought is that a closed-form description of possible CW-complex cell structure contains a ton of information about a space, but this is sort of subtle. – Badam Baplan Nov 29 '14 at 19:01
• For example, sometimes we want to ask questions about excision or quotienting that are facilitated by one cell-structure and not another at all. e.g. if we talk about homologies of $S^2$ after we quotient on an equator, we want a 1-skeleton in our description. Similar situations if we smash or wedge CW spaces in various configurations. The complete list of CW structures for a space should have information pertinent to any operation on the space we might conceive of, and I'm wondering if there is a theory that addresses the extraction of this information. – Badam Baplan Nov 29 '14 at 19:12
• Essentially, this list seems like a very algebraic description of a space and provides a natural metric for comparing how similar two CW complexes are. It contains tons of information, entirely different than homology or homotopy information which is invariant under the different cell-structures of the list. – Badam Baplan Nov 29 '14 at 19:20
• @user3499756 Hatcher provides an exercise in chapter 0 of his algebraic topology book, whose purpose I suppose is to point out this is hopeless. He asks the reader to, for any triple $(V,E,F)$ such that $V-E+F = 2$, construct a CW structure on $S^2$ with $V$ 0-cells, $E$ 1-cells, and $F$ 2-cells. (It's not particularly difficult). Indeed, there are actually lots of cell structures of $S^2$ for a given $(V,E,F)$. Also note that given two different attaching maps for a cell, if the maps are homotopy equivalent, the resulting spaces are homotopy equivalent; same replacing homotopy with isotopy. – user98602 Dec 1 '14 at 2:04
• cool, thank you, i definitely missed that exercise when i was doing chapter 0. i will check it out and ponder! – Badam Baplan Dec 1 '14 at 2:25