nLab tautological line bundle

Idea

The canonical line bundle over a projective space is sometimes called its “tautological line bundle”. (For more background see also at classifying space.)

Definition

In the following, let $k$ be a star-field, possibly a skew-field

$$k \,\in\, StarSkewFields \,.$$

In the context of algebraic topology or differential topology, one is usually interested in $k$ being one of the three associative real normed division algebras (real numbers, complex numbers or quaternions):

$$k \,\in\, \big\{ \mathbb{R}, \mathbb{C}, \mathbb{H}\big\}$$

equipped with their canonical conjugation operations.

The relevance of the quaternions here is the reason to insist on the generality of skew-fields and hence of some pedantry about order of products and distinction of left/right actions in the following. (For octonions the notion of tautological lines should make some sense over the first octonionic projective space but not beyond.)

Let $n \in \mathbb{N}$ be a natural number.

Tautological line bundle and blow-up

The tautological $k$-line bundle over the projective space $k P^n$ is the following vertical bundle map:

(1)$$\array{ \mathcal{L}_{k P^n} & \coloneqq & \frac{ (k^{n+1} \setminus \{0\}) \times k^\ast }{ k^\times } & \overset{ [v,z] \mapsto \big( [v], v \cdot z \big) }{\hookrightarrow} & \frac{ k^{n+1} \setminus \{0\} }{ k^\times } \times k^{n+1} \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P^n &=& \frac{ (k^{n+1} \setminus \{0\}) \times \ast }{ k^\times } }$$

Here:

• $k^\times \,\coloneqq\, k \setminus \{0\}$ is the group of units of $k$;

• $k^{n+1} \coloneqq \underset{n+1\;summands}{\underbrace{k \oplus \cdots \oplus k}}$ is the canonical $(n+1)$-dimensional $k$-vector space,

  whose elements we will also denote as lists

  (2)$$v \,=\, (v_1, v_{rest}) \,=\, (v_1, v_2, \cdots, v_{n+1}) \,,$$

  regarded with the right $k^\times$-group action:

  (3)$$\array{ k^{n+1} \times k^\times & \overset{ \;\;\;\; }{\longrightarrow} & k^{n+1} \\ \big( (v_1, \, \cdots, \, v_{n+1}), \, z \big) &\mapsto& (v_1 \!\cdot\! z,\, \cdots, \, v_{n+1} \!\cdot\! z) \,, }$$

• $k^\ast$ is $k$ equipped with the right $k^\times$-action by inverse multiplication from the left:

  (4)$$\array{ k^\times \times k^\ast &\longrightarrow& k^\ast \\ (z,g) &\mapsto& z^{-1} \cdot g }$$

  (equivalently, (3) and (4) are left actions of the opposite group $(k^\times)^{op}$)

• $\frac{(-) \times (-)}{k^\times}$ denotes the quotient space of a product of right $k^\times$-spaces by their diagonal action;

• $[-]$ denotes its elements as equivalence classes of elements of the original space;

• so that, for $z \,\in\, k^\times$,

  $$[v,\,z] \;=\; [v,\, z \!\cdot\! 1] \;=\; [v \!\cdot\! z ,\, 1] \;\;\in\;\; \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times }$$

• and hence so that the quotient construction in (1) is equivalently the $k$-fiber associated bundle to the $k^\times$-principal bundle (here “$T$” is for torsor):

(5)$$\array{ T_{k P^n} & \coloneqq & k^{n+1} \setminus \{0\} \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ quotient\;map }} \\ k P^n &=& (k^{n+1} \setminus \{0\}) / k^\times \,. }$$

For $k$ a star algebra, the quotient space (1) becomes a $\mathbb{K}$-vector bundle (specifically: line bundle) via the residual left action of $\mathbb{K}$ on itself by conjugate right multiplication.

The line bundle (1) is “tautological” in the sense that its fiber over a point labeled $[v]$ – which may be regarded as the name of the line spanned by the vector $v \in k^{n+1}$ – consists of all the points $v \!\cdot\! z$ on that line – as made explicit by the horizontal map in (1).

Often the tautological line bundle is referred to via the notation “$\mathcal{O}_k(-1)$”, which in algebraic geometry is standard notation for its abelian sheaf of sections (see e.g. Wirthmüller 12, p. 14 (16 of 67)):

• “$\mathcal{L}_{k P^n}$” $\;\;\;\leftrightarrow\;\;\;$ “$\mathcal{O}_k(-1)$”

The corestriction of the horizontal map in (1) to $k^{n+1}$

$$\mathcal{L}_{k P^n} \coloneqq \frac{ (k^{n+1} \setminus \{0\}) \times k^\ast }{ k^\times } \overset{ [v,z] \mapsto v \cdot z }{\longrightarrow} k^{n+1}$$

exhibits the total space of the tautological bundle as the “blow-up” of the origin of $k^{n+1}$.

The following illustration shows the tautological real line bundle over 1-dimensional real projective space, but the general picture is “the same”, up to higher dimensionality of all spaces involved:

Dual tautological line bundle and its Thom space

The dual line bundle of the tautological line bundle (1) is

(6)$$\array{ \mathcal{L}^\ast_{k P^n} & \coloneqq & \frac{ (k^{n+1} \setminus \{0\}) \times k }{ k^\times } & \overset{ [v,z] \mapsto [ (v , z) ] }{\hookrightarrow} & \frac{ k^{n+2} \setminus \{0\} }{ k^\times } & = \, k P^{n+1} \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P^n &=& \frac{ (k^{n+1} \setminus \{0\}) \times \ast }{ k^\times } }$$

with typical fiber $k$ instead of $k^\ast$, meaning that the action of $k^\times$ on the fibers is now the direct right multiplication action (3), instead of the dual action (4).

Often the dual tautological line bundle is referred to via the notation “$\mathcal{O}_k(1)$”, which in algebraic geometry is standard notation for its abelian sheaf of sections:

• “$\mathcal{L}^\ast_{k P^n}$” $\;\;\;\leftrightarrow\;\;\;$ “$\mathcal{O}_k(1)$”.

The horizontal map in (6) embeds the complement of the single point $[(v\! = \! 0,\,1)] \in k P^{n+1}$. That point however is the limit as $z \to \infty$, hence is the image of the base point as (6) extends to a map on the Thom space of the dual tautological line bundle:

(7)$$Th \left( \mathcal{L}^\ast_{k P^n} \right) \underoverset {\simeq} { [v,z] \mapsto \left\{ \array{ [(0,1)] &\vert& z = \infty \\ [ (v , z) ] &\vert& else } \right. }{\longrightarrow} k P^{n+1} \,.$$

(E.g.: Tamaki & Kono 2006, Part III, Lemma 3.8, bewaring that these authors secretly identify lines with dual lines – as seen from the inner product used in the second but last line on their p. 46. Also see Conner & Floyd 1966, Part I, Prop. 4.3 for an alternative perspective in terms of coset spaces.)

Notice how the inclusion of the point at $[v,\, z\!=\!\infty] \leftrightarrow [(v\!=\!0,\,1)]$ in the Thom space interplays with the condition that $v = 0$ is excluded in the base space.

Under the canonical inclusion of projective spaces $k P^n \hookrightarrow k P^{n+1}$ their dual tautological line bundles (6) evidently pullback to each other, and their total spaces compatibly include into each other:

$$\array{ [v,z] &\mapsto& [(0,v),z] \\ \mathcal{L}^\ast_{k P^n} & \overset{\;\;\;\;\;\;}{\hookrightarrow} & \mathcal{L}^\ast_{k P^{n+1}} \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ k P^n & \overset{\;\;\;\;\;\;}{\hookrightarrow} & k P^{n+1} & \\ [v] &\mapsto& [(0,v)] \,. }$$

Here the coordinate expressions make manifest that the induced inclusions of the Thom spaces of the tautological line bundles recover, under the identification (7), the canonical inclusion of the projective spaces:

(8)$$\array{ [(v,z)] &\mapsto& [v,z] &\mapsto& [(0,v),z] &\mapsto& [(0,v,z)] \\ k P^{n+1} &=& Th \big( \mathcal{L}^\ast_{k P^n} \big) & \overset{\;\;\;\;\;\;}{\hookrightarrow} & Th \big( \mathcal{L}^\ast_{k P^{n+1}} \big) &=& k P^{n+2} \\ && {}^{\mathllap{zero \atop section}} \big\uparrow && \big\uparrow {}^{\mathrlap{zero \atop section}} \\ k P^n &=& k P^n & \overset{\;\;\;\;\;\;}{\hookrightarrow} & k P^{n+1} &=& k P^{n+1} \\ [v] &\mapsto& [v] &\mapsto& [(0,v)] &\mapsto& [(0,v)] \,. }$$

Notice how, in this coordinatization, the projective spaces are horizontally included by adjoining a 0-coordinate to the left of the list (2) and vertically by adjoining a 0-coordinate to the right.

It follows that under forming a suitable colimit over this diagram as $n \to \infty$, in a suitable category (typically in homotopy types of topological spaces if $k$ is a topological field, see also below), the infinite projective space wants to be equivalent to the Thom space of its dual tautological line bundle:

$$k P^\infty \;\simeq\; k P^{\infty + 1} \;\simeq\; Th \big( \mathcal{L}^\ast_{k P^\infty} \big) \,.$$

See at zero-section into Thom space of universal line bundle is weak equivalence.

As a topological line bundle

We make fully explicit how the tautological line bundle (1) is a locally trivial topological vector bundle. Hence regard now $k$ as a topological field, either

• $k = \mathbb{R}$ the real numbers,

• or $k = \mathbb{C}$ the complex numbers

equipped with their Euclidean metric topology.

Definition

(topological projective space)

Let $n \in \mathbb{N}$. Consider the Euclidean space $k^{n+1}$ equipped with its metric topology, let $k^{n+1} \setminus \{0\} \subset k^{n+1}$ be the topological subspace which is the complement of the origin, and consider on its underlying set the equivalence relation which identifies two points if they differ by multiplication with some $c \in k$ (necessarily non-zero):

$$(\vec x_1 \sim \vec x_2) \;\Leftrightarrow\; \left( \underset{c \in k}{\exists} ( \vec x_2 = c \vec x_1 ) \right) \,.$$

The equivalence class $[\vec x]$ is traditionally denoted

$$[x_1 : x_2 : \cdots : x_{n+1}] \,.$$

Then the projective space $k P^n$ is the corresponding quotient topological space

$$k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim \,.$$

Definition

(standard open cover of topological projective space)

For $n \in \mathbb{N}$ the standard open cover of the projective space $k P^n$ (def. ) is

$$\left\{ U_i \subset k P^n \right\}_{i \in \{1, \cdots, n+1\}}$$

with

$$U_i \coloneqq \left\{ [x_1 : \cdots : x_{n+1}] \in k P^n \;\vert\; x_i \neq 0 \right\} \,.$$

To see that this is an open cover:

1. This is a cover because with the orgin removed in $k^n \setminus \{0\}$ at every point $[x_1: \cdots : x_{n+1}]$ at least one of the $x_i$ has to be non-vanishing.

2. These subsets are open in the quotient topology $k P^n = (k^{n+1} \setminus \{0\})/\sim$, since their pre-image under the quotient co-projection $k^{n+1} \setminus \{0\} \to k P^n$ coincides with the pre-image $(pr_i\circ\iota)^{-1}( k \setminus \{0\} )$ under the projection onto the $i$th coordinate in the product topological space $k^{n+1} = \underset{i \in \{1,\cdots, n+1\}}{\prod} k$ (where we write $k^{n+1} \setminus \{0\} \overset{\iota}{\hookrightarrow} k^{n+1} \overset{pr_i}{\to} k$).

Definition

(tautological topological line bundle)

For $k$ a topological field and $n \in \mathbb{N}$, the tautological line bundle over the projective space $k P^n$ is the topological $k$-line bundle whose total space is the following subspace of the product space of the projective space $k P^n$ with $k^{n+1}$:

$$T \coloneqq \left\{ ( [x_1: \cdots : x_{n+1}], \vec v) \in k P^n \times k^{n+1} \;\vert\; \vec v \in \langle \vec x\rangle_k \right\} \,,$$

where $\langle \vec x\rangle_k \subset k^{n+1}$ is the $k$-linear span of $\vec x$.

(The space $T$ is the space of pairs consisting of the “name” of a $k$-line in $k^{n+1}$ together with an element of that $k$-line).

This is a bundle over projective space via the projection function

$$\array{ T &\overset{\pi}{\longrightarrow}& k P^n \\ ([x_1: \cdots : x_{n+1}], \vec v) &\mapsto& [x_1: \cdots : x_{n+1}] } \,.$$

Proposition

(tautological topological line bundle is well defined)

The tautological line bundle in def. is well defined in that it indeed admits a local trivialization.

Proof

We claim that there is a local trivialization over the canonical open cover from def. . This is given for $i \in \{1, \cdots, n\}$ by

$$\array{ U_i \times k &\overset{}{\longrightarrow}& T\vert_{U_i} \\ ( [x_1 : \cdots x_{i-1}: 1 : x_{i+1} : \cdots : x_{n+1}] , c ) &\mapsto& ( [x_1 : \cdots x_{i-1} : 1 : x_{i+1} : \cdots : x_{n+1} ], (c x_1, c x_2, \cdots , c x_{n+1}) ) } \,.$$

First, this is clearly a bijection of underlying sets.

To see that this function and its inverse function are continuous, hence that this is a homeomorphism notice that this map is the extension to the quotient topological space of the analogous map

$$\array{ ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , c) &\mapsto& ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , (c x_1, \cdots c x_{i-1}, c, c x_{i+1}, \cdots, c x_{n+1}) ) } \,.$$

This is a polynomial function on Euclidean space and since polynomials are continuous it is continuous. Similarly, the inverse function lifts to a rational function on a subspace of Euclidean space, and since rational functions are continuous on their domain of definition, also this lift is continuous.

Therefore by the universal property of the quotient topology, also the original functions are continuous.

Examples

Möbius strip

The tautological line bundle over the 1-dimensional real projective space $\mathbb{R}P^1$ is the Möbius strip.

Over Riemann sphere

The basic complex line bundle on the 2-sphere is the tautological complex line bundle over the complex projective space $\mathbb{C}P^1 \simeq S^2$ (the Riemann sphere).

This plays a key role in topological K-theory and more generally in complex oriented cohomology theory.

Related concepts

• canonical line bundle, universal complex line bundle

• hyperplane line bundle

• tautological equivariant line bundle

• resolution of singularities

• Bott element

References

Discussion with an eye towards complex-oriented cohomology theory:

• Pierre Conner, Edwin Floyd, p. 21 (28 of 120) in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Dai Tamaki, Akira Kono, Chapter 3, Section 2 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (pdf, ISBN: 978-0-8218-3514-2)

Lecture notes with an eye towards topological K-theory:

• Klaus Wirthmüller, section 2 of Vector bundles and K-theory, 2012 (pdf)

See also:

• Wikipedia, Tautological bundle