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cell structure of projective spaces

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Complex geometry

Contents

Idea

The projective spaces over ground ring 𝕂\mathbb{K} being the real numbers ℝ\mathbb{R}, complex numbers β„‚\mathbb{C}, quaternions ℍ\mathbb{H} or octonions 𝕆\mathbb{O} admit, respectively, CW--cell complex structure with a single cell in each dimension kk, 2k2 k, 4k4 k or 8k8k, respectively.

We discuss the cases π•‚βˆˆ{ℝ,β„‚,ℍ}\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}. For the case 𝕂=𝕆\mathbb{K} \,=\, \mathbb{O} see Lackman 19, Lemma 3.4

Preliminaries

Let π•‚βˆˆ{ℝ,β„‚,ℍ}\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 ℝ dim ℝ(𝕂)\mathbb{R}^{ dim_{_{\mathbb{R}}}(\mathbb{K}) }.

For nβˆˆβ„•n \in \mathbb{N}, consider

𝕂P n≔(𝕂 n+1βˆ–{0})/𝕂 Γ— \mathbb{K}P^n \;\; \coloneqq \;\; \big( \mathbb{K}^{n+1} \setminus \{0\} \big) / \mathbb{K}^\times

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

Notice the canonical subspace inclusion

𝕂P n β†ͺ 𝕂P n+1 [z 0:β‹―:z n] ↦ [z 0:β‹―:z n:0]. \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 𝕂P n+1βˆ–π•‚P\mathbb{K}P^{n+1} \setminus \mathbb{K}P with topological boundary

βˆ‚(𝕂P n+1βˆ–π•‚P)≔(𝕂P n+1βˆ–π•‚P)Β―βˆ–(𝕂P n+1βˆ–π•‚P). \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 𝕂\mathbb{K}-projective space)

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

D (n+1)β‹…dim ℝ(𝕂) ⟢ 𝕂P n+1 ↑ (po) ↑ S (n+1)β‹…dim ℝ(𝕂)βˆ’1 ⟢ 𝕂P n \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 𝕂P n+1\mathbb{K}P^{n+1} as the result of an (n+1)β‹…dim ℝ(𝕂)(n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K})-cell attachment to 𝕂P n\mathbb{K}P^{n} with attaching map the canonical projection

S (n+1)β‹…dim ℝ(𝕂)βˆ’1≃(𝕂 n+1βˆ–{0})/ℝ + Γ— ⟢ (𝕂 n+1βˆ–{0})/𝕂 ×≃𝕂P n. \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)𝕂P n β†ͺ 𝕂P n+1 [z 0:β‹―:z n] ↦ [z 0:β‹―:z n:0] \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

𝕂P n+1βˆ–π•‚P n={[z 0:β‹―:z n:z n+1]|z n+1β‰ 0}βŠ‚π•‚P n+1. \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)𝕂P n+1βˆ–π•‚P n βŸΆβ‰ƒ Int(D (n+1)dim ℝ(𝕂))≃{(y 0,β‹―,y n,(1βˆ’r))|r∈[0,1)βŠ‚β„, |yβ†’| 2+(1βˆ’r) 2=1} βŠ‚π•‚ n+2 [z 0:β‹―:z n:z n+1] ↦ 1|zβ†’|(z 0β‹…z n+1 *|z n+1|,β‹―,z nβ‹…z n+1 *|z n+1|,|z n+1|). \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→1r \to 1 on the right, we see that the inverse of this homeomorphism gives horizontal isomorphisms in a commuting square of the following form:

D (n+1)β‹…dim ℝ(𝕂) βŸΆβ‰ƒ 𝕂P n+1βˆ–π•‚P nΒ― ↑ ↑ S (n+1)β‹…dim ℝ(𝕂)βˆ’1 βŸΆβ‰ƒ βˆ‚(𝕂P n+1βˆ–π•‚P n). \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 𝕂P n+1\mathbb{K}P^{n+1} is manifestly the union of its subspace 𝕂P n\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:

D (n+1)β‹…dim ℝ(𝕂) βŸΆβ‰ƒ 𝕂P n+1βˆ–π•‚P nΒ― ⟢ 𝕂P n+1 ↑ ↑ (po) ↑ S (n+1)β‹…dim ℝ(𝕂)βˆ’1 βŸΆβ‰ƒ βˆ‚(𝕂P n+1βˆ–π•‚P n) ⟢ 𝕂P n \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 𝕂\mathbb{K}-projective spaces)

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

*≃𝕂P 0β†ͺAAA𝕂P 1β†ͺAAA𝕂P 2β†ͺAAAβ‹―β†ͺAAA𝕂P ∞ \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 nn exhibit CW-complex-structure on 𝕂P n\mathbb{K}P^n with single cells in degree kβ‹…dim ℝ(𝕂)k \cdot dim_{\mathbb{R}}(\mathbb{K}).

Consequences

Fixing an orthonormal basis i,j,kβˆˆβ„\mathrm{i}, \mathrm{j}, \mathrm{k} \,\in\, \mathbb{H} of imaginary quaternions with iβ‹…j=k\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 ℍ\mathbb{H} as a β„‚\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 β„β‰ƒβ„‚βŠ•β„‚ *\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 kβˆˆβ„•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 β€œtwistor fibration” for k=1k = 1);

  2. the bottom right vertical morphism is the canonical inclusion β„‚P 2k+2β†ͺβ„‚P 2k+3\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=1k = 1:

Here the notation closely alludes to the construction inside the proof of Prop. : In particular 1βˆ’r1-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 β€œto themselves”, and, if necessary, by including a last coordinate 1βˆ’r=01 - r = 0.

References

The case of octonionic projective space:

Last revised on January 30, 2021 at 04:26:29. See the history of this page for a list of all contributions to it.