Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 ?

share|improve this question
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
show 1 more comment

1 Answer

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.