What is $\mathbb{P}^{\infty}$?

Question

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it?

I know $\mathbb{P}^{n}$ is a space of $1-$dimensional subspaces of $\mathbb{C}^{n+1}$, but have never seen $\mathbb{P}^{\infty}$.

Answer 1

There is a natural map $\varphi_n : \mathbb{CP}^n \to \mathbb{CP}^{n+1}$ given by $\varphi_n([z_0, \dots, z_n]) = [z_0, \dots, z_n, 0]$. This defines a direct system and therefore has a direct limit denoted $\mathbb{CP}^{\infty}$.

More concretely, an element of $\mathbb{CP}^{\infty}$ can be written as $[z_0, z_1, z_2, \dots]$ where each component $z_i$ is a complex number, at least one of which is non-zero, and all but finitely many of which are zero. As in the finite dimensional case, $[\lambda z_0, \lambda z_1, \lambda z_2, \dots] = [z_0, z_1, z_2, \dots]$ for all $\lambda \in \mathbb{C}^*$.

The same construction can be used to produce $\mathbb{C}^{\infty}$ which is an infinite-dimensional vector space. The space of one-dimensional subspaces of $\mathbb{C}^{\infty}$ is precisely $\mathbb{CP}^{\infty}$.

From a more topological viewpoint, $\mathbb{CP}^n$ has a CW structure consisting of a single cell in every even dimension between $0$ and $2n$ inclusive and no other cells, i.e. $\mathbb{CP}^n = e^0\cup e^2\cup\dots\cup e^{2n}$. The map $\varphi_n$ identifies the CW complex of $\mathbb{CP}^n$ as a CW subcomplex of $\mathbb{CP}^{n+1}$. Therefore, one can view $\mathbb{CP}^{n+1}$ as a space obtained by attaching a $(2n+2)$-cell to $\mathbb{CP}^n$. Continuing this process indefinitely, we obtain a CW complex $\mathbb{CP}^{\infty}$ which consists of a single cell in every even dimension and no other cells, i.e. $\mathbb{CP}^{\infty} = e^0\cup e^2\cup\dots$

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Note, we can obtain other spaces in a similar fashion, such as $S^{\infty}, \mathbb{RP}^{\infty}$, and $\mathbb{HP}^{\infty}$. Such spaces play very important roles in algebraic topology as they are examples of classifying spaces and Eilenberg-MacLane spaces.

• $\mathbb{RP}^{\infty}$ is the classifying space for $O(1) \cong \mathbb{Z}/2\mathbb{Z}$ and is a $K(\mathbb{Z}/2\mathbb{Z}, 1)$,
• $\mathbb{CP}^{\infty}$ is the classifying space for $U(1) \cong SO(2)$ and is a $K(\mathbb{Z}, 2)$,
• $\mathbb{HP}^{\infty}$ is the classifying space for $\operatorname{Spin}(3) \cong Sp(1) \cong SU(2)$ and is not an Eilenberg-Maclane space as discussed here,
• $S^{\infty}$ is the total space of the universal bundles on $\mathbb{RP}^{\infty}$, $\mathbb{CP}^{\infty}$, and $\mathbb{HP}^{\infty}$.

Note that $O(1)$, $U(1)$, and $Sp(1)$ give the spheres $S^0$, $S^1$, and $S^3$ respectively the structure of a topological group. These are the only spheres which admit such a structure.