# Removing cells from CW complex and without changing the homotopy type

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of spaces which are homotopy equivalent to finite CW complexes.

It is known that one CW complex structure for the projective space $\mathbb{R}P^n$ consists of one cell in each dimension and the attaching maps being the projection maps from $\mathbb{S}^n$ to $\mathbb{R}P^n$ (See Hatcher Example 0.4).

My first question is:

Could we delete some of the cells of that CW complex structure without changing the homotopy type?

Assuming that the answer was negative, this lead us to my second question:

Could we "play"(add cells and remove others) with the cells in order to obtain a another CW complex homotopy equivalent to the first one but without cells in all dimensions?

Thanks in advance and any help would be appreciated.

• here you want answer for the particular case of $\mathbb RP^n$ or in general ? – Anubhav Mukherjee Feb 13 '16 at 15:43
• I wanted a particular answer but Michael Albanese's answer in math.stackexchange.com/questions/1653312/… has solved my question. Many thanks anyway @Anubhav.K! – D1811994 Feb 13 '16 at 15:50
• There is a small flaw in his answer – Anubhav Mukherjee Feb 13 '16 at 15:53
• I should point out that the flaw that Anubhav.K is referring to has since been fixed. – Michael Albanese Mar 13 '16 at 20:13

Michael Albanese gave me an answer in CW complex structure for $$\mathbb{R}P^n$$ or a space homotopy equivalent which is applicable here to the particular case of the projective space which is the want I cared about.