Given a finite connected CW complex $X$, one can asked 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 ?