In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can anyone provide a reference or a proof?



Why is complex projective space triangulable?
I will present a triangulation of $\mathbb{CP}^{n-1}$. More specifically, I will give an explicit regular CW structure on $\mathbb{CP}^{n-1}$. As spinorbundle says, the first barycentric subdivision of a regular CW complex is a simplicial complex homeomorphic to the original CW complex.




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Recall that to put a regular CW complex on space $X$ means to decompose $X$ into disjoint pieces $Y\_i$ such that:


(1) The closure of each $Y\_i$ is a union of $Y$'s.


(2) For each $i$, the pair $(\overline{Y\_i}. Y\_i)$ is homemorphic to $(\mbox{closed}\ d-\mbox{ball}, \mbox{interior of that}\ d-\mbox{ball})$ for some $d$.


The barycentric subdivision of $X$ corresponding to this regular CW complex is the simplicial complex which has a vertex for each $Y\_i$ and has a simplex $(i\_0, i\_1, \ldots, i\_r)$ if and only if $\overline{Y\_{i\_0}} \subset \overline{Y\_{i\_1}} \subset \cdots \subset \overline{Y\_{i\_r}}$.




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Write $(t\_1: t\_2: \ldots: t\_n)$ for the homogeneous coordinates on $\mathbb{CP}^{n-1}$. For $I$ a nonempty subset of $\{ 1,2, \ldots, n \}$, let $Z\_I$ be the subset of $\mathbb{CP}^{n-1}$ where $|t\_i|=|t\_{i'}|$ for $i$ and $i' \in I$ and $|t\_i| > |t\_j|$ for $i \in I$ and $j \not \in I$. Note that $Z\_I \cong (S^1)^{|I|-1} \times D^{2(n-|I|)}$, where $D^k$ is the open $k$-disc. Also, $\overline{Z\_I} = \bigcup\_{J \supseteq I} Z\_J \cong (S^1)^{|I|-1} \times \overline{D}^{2(n-|I|)}$ where $\overline{D}^k$ is the closed $k$-disc.


We now cut those torii into discs. For $i$ and $i'$ in $I$, cut $Z\_I$ along $t\_i=t\_{i'}$ and $t\_i = - t\_{i'}$. So the combinatorial data indexing a face of this subdivision is a cyclic arrangement of the symbols $i$ and $-i$, for $i \in I$, with $i$ and $-i$ antipodal to each other. For example, let $I=\{ 1,2,3,4,5 \}$ and write $t\_k=e^{i \theta\_k}$ for $k \in I$. Then one of our faces corresponds to the situation that, cyclically, 
$$\theta\_1 < \theta\_2 = \theta\_4 + \pi < \theta\_3 = \theta\_5 < \theta\_1+ \pi < \theta\_2 + \pi = \theta\_4 < \theta\_3 + \pi = \theta\_5 + \pi < \theta\_1.$$
This cell is clearly homeomorphic to $\{ (\alpha, \beta) : 0 < \alpha < \beta < \pi \}$. Similarly, each of these cells is an open ball, and each of their closures is a closed ball. We have put a CW structure on the torus.


Cross this subdivision of the torus with the open disc $D^{2(n-|I|)}$. The result, if I am not confused, is a regular $CW$ decomposition of $\mathbb{CP}^{n-1}$.


