# Minimal number of cells of a CW complex (up to homotopy)

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 ?

• 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 Commented Oct 24, 2012 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. Commented Oct 24, 2012 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. Commented Oct 25, 2012 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. Commented Oct 25, 2012 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! Commented Oct 25, 2012 at 1:56