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Given a finite connected CW complex $X$, one can ask what can said about the number of its cells.

As an example, let's estimate the number of 1-cells: There is an epimorphism $\pi_1(X_1) \to \pi_1(X)$ from the 1-skeleton which is a connected graph and whose fundamental group is a free group on a subset of the 1-cells of $X$ [Hatcher, 1A.2]. Hence we have the lower bound $$\text{number of 1-cells } \ge \text{ minimal number of generators of } \pi_1(X)$$ Conversely, given a presentation of $\pi_1(X)$ with a minimal number of generators $d$, there is a CW complex $X'$ with $d$ 1-cells and $\pi_1(X') = \pi_1(X)$ [Hatcher, 1.28].

Question 1: Can $X'$ be choosen to be (cellularly) homotopy equivalent to $X$ ?

Futhermore, by taking into account the cellular chain complex, it's not hard to see that the number of $n$-cells $(n \ge 0)$ is bounded below by the fact that we need (at least)

  • one $n$-cell for each direct summand of $H_n(X)$
  • one $n$-cell for each direct summand of finite order of $H_{n-1}(X)$

As formula: $$\text{number of n-cells } \ge d(H_nX) + d(H_{n-1}(X)_{tor}) =: m_n(X)\qquad(\ast)$$ where $d(\cdot)$ denotes the minimal number of generators.

It's known that if $X$ is simply connected, then $X$ is homotpoy equivalent to a complex $X'$ having exactly $m_n(X)$ cells in each dimension [Hatcher, 4C.1].

Question 2: Are there other classes of CW complexes where each $X$ is homotopy equivalent to a complex $X'$ that has $m_n(X)$ cells in each dimension ?

By the estimate above, a necessary condition for such a class is $d(\pi_1X)=d(\pi_1(X)_{ab})$, e.g. $\pi_1(X)$ solvable.

Question 3: What's the best current bound for the minimal number of cells of (not necessarily simply-connected) finite CW complexes ?

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The tool to find minimal CW-decompositions of a space (up to homotopy-equivalence) is called Whitehead's Theorem. It's covered in Hatcher's textbook, at the start of Chapter 4.1. So Q1 has an affirmative answer. Q2 has a negative answer (think of knot complements). Q3 yes, of course. Think about the homology of covering spaces as modules over the group of covering transformations, for example –  Ryan Budney Oct 24 '12 at 23:23
    
@Ryan: Thanks for your comment and answer. Q2: I thought spaces with abelian fundamental group or classifying spaces might be candidates. I don't understand why your argument ("think of knot complements") completely sorts out these classes. Q3: Can you give some more details on how the improved bound looks like. –  Ralph Oct 24 '12 at 23:54
    
The complement of a knot in the 3-sphere has the same homology as $S^1$. For example, the degree of the Alexander polynomial is the rank of $H_1$ of the universal abelian cover of the knot complement (with rational coefficients). The number of 1-cells must therefore be at least as large as the degree of the Alexander polynomial. –  Ryan Budney Oct 25 '12 at 0:03
    
Thanks for the explanation. However, Q2 doesn't ask for particular counterexamples but for interesting classes of spaces where $(\ast)$ gives the minimal number of cells. –  Ralph Oct 25 '12 at 0:22
    
@Ryan: is it obvious when the Whitehead group of $\pi_1 X$ is trivial that indeed such an $X'$ exists? Thanks for your help understanding this! –  Patricia Hersh Oct 25 '12 at 1:56

1 Answer 1

To expand on my comment, there's a very general tool to manipulate CW-complexes, due to Whitehead. It tells you when you can in effect remove a cell from a CW-decomposition via `elementary moves', usually called Whitehead Moves. In smooth manifold theory there are parallel constructions -- people talk about "handle slides" and "handle cancellations". This comes up in the proof of the h and s-cobordism theorems, which are the smooth-category analogue of the Whitehead moves. Technically these moves have to do with a slightly more refined notion of homotopy-equivalence, called simple homotopy equivalence. Provided the fundamental group of the CW-complex is trivial, simple homotopy-equivalences are in effect the same as homotopy-equivalances, but in general they're a little more fussy.

What are the Whitehead moves? On the 0-skeleton, it's the move where you collapse a maximal forest in the 1-skeleton. On the 1-skeleton these are moves where you can cancel a 1-cell using a 2-cell that's incident to it only once. This is explained in detail in Marshall Cohen's "A course in simple-homotopy theory". GTM 10 Springer-Verlag.

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