nLab cell structure of projective spaces

Idea

The projective spaces over ground ring $\mathbb{K}$ being the real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, quaternions $\mathbb{H}$ or octonions $\mathbb{O}$ admit, respectively, CW--cell complex structure with a single cell in each dimension $k$, $2 k$, $4 k$ or $8k$, respectively.

We discuss the cases $\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$. For the case $\mathbb{K} \,=\, \mathbb{O}$ see Lackman 19, Lemma 3.4

Preliminaries

Let $\mathbb{K} \,\in\, \big\{ \mathbb{R}, \mathbb{C}, \mathbb{H} \big\}$ be one of the associative real normed division algebras, hence either the real numbers, or the complex numbers or the quaternions. We regard this as a topological ring with respect to the canonical underlying topology of the Euclidean space $\mathbb{R}^{ dim_{_{\mathbb{R}}}(\mathbb{K}) }$.

For $n \in \mathbb{N}$, consider

$$\mathbb{K}P^n \;\; \coloneqq \;\; \big( \mathbb{K}^{n+1} \setminus \{0\} \big) / \mathbb{K}^\times$$

the $\mathbb{K}$-projective space, i.e. the topological quotient space of the complement of zero in the $n+1$-fold Cartesian product of $\mathbb{K}$ by the right (say) multiplication action by the group of units $\mathbb{K}^\times \,=\, \mathbb{K} \setminus \{0\}$.

Notice the canonical subspace inclusion

$$\array{ \mathbb{K}P^n &\hookrightarrow& \mathbb{K}P^{n+1} \\ \big[ z_0 \,:\, \cdots \,:\, z_n \big] &\mapsto& \big[ z_0 \,:\, \cdots \,:\, z_n \,:\, 0 \big] \,. }$$

with induced complement $\mathbb{K}P^{n+1} \setminus \mathbb{K}P$ with topological boundary

$$\partial \big( \mathbb{K}P^{n+1} \setminus \mathbb{K}P \big) \;\; \coloneqq \;\; \overline{ \big( \mathbb{K}P^{n+1} \setminus \mathbb{K}P \big) } \setminus \big( \mathbb{K}P^{n+1} \setminus \mathbb{K}P \big) \,.$$

Statement

Proposition

(cell structure of $\mathbb{K}$-projective space)

We have a pushout diagram in in topological spaces of the form

$$\array{ D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) } &\longrightarrow& \mathbb{K}P^{n+1} \\ \big\uparrow &{}_{^{(po)}}& \big\uparrow \\ S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 } &\longrightarrow& \mathbb{K}P^n }$$

exhibiting $\mathbb{K}P^{n+1}$ as the result of an $(n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K})$-cell attachment to $\mathbb{K}P^{n}$ with attaching map the canonical projection

$$\array{ S^{(n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K})-1} \;\simeq\; \big( \mathbb{K}^{n+1} \setminus \{0\} \big) / \mathbb{R}_+^\times &\longrightarrow& \big( \mathbb{K}^{n+1} \setminus \{0\} \big) / \mathbb{K}^\times \;\simeq\; \mathbb{K}P^n \,. }$$

Proof

If we coordinatize the inclusion of consecutive projective spaces as

(1)$$\array{ \mathbb{K}P^n &\overset{\;\;\;}{\hookrightarrow}& \mathbb{K}P^{n+1} \\ [z_0 \colon \cdots \colon z_n ] &\mapsto& [z_0 \colon \cdots \colon z_n \colon 0] }$$

then the complement of this inclusion inherits coordinatization as

$$\mathbb{K}P^{n+1} \setminus \mathbb{K}P^{n} \;=\; \big\{ \left. [z_0 \colon \cdots \colon z_n \colon z_{n + 1}] \,\right\vert\, z_{n + 1} \neq 0 \big\} \;\subset\; \mathbb{K}P^{n+1} \,.$$

In terms of these coordinates, observe the following homeomorphism:

(2)$$\array{ \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n &\overset{\;\;\;\simeq\;\;\;}{\longrightarrow}& Int \big( D^{(n+1) dim_{{}_{\mathbb{R}}}(\mathbb{K})} \big) \,\simeq\, \left\{ \big( y_0, \cdots, y_n, (1-r) \big) \,\left\vert\, \array{ r \in [0,1) \subset \mathbb{R}\,, \\ \left\vert \vec y \right\vert^2 + (1-r)^2 = 1 } \right. \right\} & \subset \mathbb{K}^{n+2} \\ \big[ z_0 \,\colon\, \cdots \,\colon\, z_n \,\colon\, z_{n+1} \big] &\mapsto& \tfrac{1}{\left\vert \vec z\right\vert} \Big( z_0 \cdot \tfrac{ z^\ast_{n+1} }{ \left\vert z_{n+1}\right\vert } \,,\, \cdots \,,\, z_n \cdot \tfrac{ z^\ast_{n+1} }{ \left\vert z_{n+1}\right\vert } \,,\, \left\vert z_{n+1}\right\vert \Big) \,. }$$

In words this says: Given the class of a set of homogeneous coordinates with the last one non-zero, form the unique representative vector subject to the condition that:

1. its last coordinate is real;

2. its norm is unity.

Noticing that (2) takes the topological boundary on the left to the boundary sphere as $r \to 1$ on the right, we see that the inverse of this homeomorphism gives horizontal isomorphisms in a commuting square of the following form:

$$\array{ D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) } &\overset{\;\;\simeq\;\;}{\longrightarrow}& \overline{ \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n } \\ \big\uparrow && \big\uparrow \\ S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 } &\overset{\;\;\simeq\;\;}{\longrightarrow}& \partial \big( \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n \big) \,. }$$

But since $\mathbb{K}P^{n+1}$ is manifestly the union of its subspace $\mathbb{K}P^n$ with the topological closure of the complement of this subspace, we have a pushout square as on the right of the following pasting diagram:

$$\array{ D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) } &\overset{\;\;\simeq\;\;}{\longrightarrow}& \overline{ \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n } &\longrightarrow& \mathbb{K}P^{n+1} \\ \big\uparrow && \big\uparrow &{}_{^{(po)}}& \big\uparrow \\ S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 } &\overset{\;\;\simeq\;\;}{\longrightarrow}& \partial \big( \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n \big) &\longrightarrow& \mathbb{K}P^n }$$

It follows that the total rectangle is a pushout (if you wish: by the pasting law, using that all commuting squares with parallel isomorphisms are pushouts).

Corollary

(CW-complex-structure on $\mathbb{K}$-projective spaces)

The $\mathbb{K}$-projective spaces appear in cotowers

$$\ast \,\simeq\, \mathbb{K}P^0 \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{K}P^1 \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{K}P^2 \overset{\phantom{AAA}}{\hookrightarrow} \cdots \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{K}P^\infty$$

which at each stage $n$ exhibit CW-complex-structure on $\mathbb{K}P^n$ with single cells in degree $k \cdot dim_{\mathbb{R}}(\mathbb{K})$.

Consequences

Fixing an orthonormal basis $\mathrm{i}, \mathrm{j}, \mathrm{k} \,\in\, \mathbb{H}$ of imaginary quaternions with $\mathrm{i} \cdot \mathrm{j} \,=\, \mathrm{k}$ induces, in particular, a star-algebra inclusion of the complex numbers into the quaternions

and a direct sum decomposition of $\mathbb{H}$ as a $\mathbb{C}$-bimodule, with right action by direct right multiplication but left action by complex conjugate left multiplication in the second variable:

Proposition

Under a decomposition $\mathbb{H} \,\simeq\, \mathbb{C} \oplus \mathbb{C}^\ast$ as above, the cell structures on complex projective spaces and quaternionic projective spaces from Prop. are compatible, in that for all $k \in \mathbb{N}$ we have a pasting diagram of the form

where the top square and the total rectangle are the pushout-squares from Prop. , while in the bottom square

1. the bottom left vertical morphism is the canonical projection (the “twistor fibration” for $k = 1$);

2. the bottom right vertical morphism is the canonical inclusion $\mathbb{C}P^{2k + 2} \hookrightarrow \mathbb{C}P^{2k+3}$ (1) followed by that canonical projection.

In particular, it follows by the pasting law that this bottom square is also a pushout.

Proof

First, the bottom left morphism clearly has to be the claimed projection for the total left morphism to be the assumed projection.

Next, by the universal property of the top pushout, the bottom right morphism is unique once it is such as to induce the given total rectangle. So we just have to check that the total right vertical morphism factors as claimed. This is a straightforward unwinding of the construction of these morphisms in the proof of Prop. .

The following diagram means to make this evident for the case that $k = 1$:

Here the notation closely alludes to the construction inside the proof of Prop. : In particular $1-r$ denotes a real number (regarded inside the complex numbers or quaternions under the chosen embedding above), using that there is always a homogeneous coordinate representative with the last entry of this form. With this understood, the maps are given by sending coordinate labels “to themselves”, and, if necessary, by including a last coordinate $1 - r = 0$.

Related statements

• zero-section into Thom space of universal line bundle is weak equivalence

References

The case of octonionic projective space:

• Malte Lackmann, Lemma 3.4 in: The octonionic projective plane (arXiv:1909.07047)