Similarly, I inclined to complain that it is not the vector bundle itself that splits when pulled back to its own projective bundle, not naturally, but its dual bundle.

I think I will edit accordingly at *projective bundle*. Once I have more wifi…

Thanks.

By the way, it’s remarkable how most of the literature glosses over the one non-trivial point of the whole discussion: the distiction between the roles of the tautological bundle and its dual.

It’s the *dual* tautological bundle whose Thom space is *naturally* identified with the next projective space.

The notation that I chose in the entry is meant to make this manifest.

Compare to Tamaki-Kono 06, Part III, Lemma 3.8, who stand out as it not being beneath them to write the proof: Their lemma talks about the non-dual line bundle, but then in their proof a choice of isomorphism of the typical fiber with its dual is silently introduced (their dot product, second but last line of the page).

]]>Fixed minor typo:

$k^{n+1} \coloneqq \underset{n+1\;summands}{\underbrace{k \oplus \cdots \oplus k}}$It had “$k\;summands$”.

]]>I have fixed one typo in the ordering of factors, and took that as occasion to expand a bit more on the fine-print of ordering factors in the formulas, for the case over skew-fields.

]]>I have added remark on the equivalence $k P^{n + 1} \simeq Th\big( \mathcal{L}_{k P ^n}\big)$ as $n$ ranges.

Currently it reads like so:

Under the canonical inclusion of projective spaces $k P^n \hookrightarrow k P^{n+1}$ their dual tautological line bundles (eq:DualTautologicalBundleProjection) 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 (eq:ThomSpaceOfDualTautologicalBundleIsNextProjectiveSpace), the canonical inclusion of the projective spaces:

$\array{ [(v,z)] &\mapsto& [v,z] &\mapsto& [(0,v),z] &\mapsto& [(0,v,z)] \\ k P^{n+1} &=& Th \big( \mathcal{L}^\ast_{k P^n} \big) & \overset{\;\;\;\;\;\;}{\hookrightarrow} & Th \big( \mathcal{L}^\ast_{k P^{n+1}} \big) &=& k P^{n+2} \\ && {}^{\mathllap{zero \atop section}} \big\uparrow && \big\uparrow {}^{\mathrlap{zero \atop section}} \\ k P^n &=& k P^n & \overset{\;\;\;\;\;\;}{\hookrightarrow} & k P^{n+1} &=& k P^{n+1} \\ [v] &\mapsto& [v] &\mapsto& [(0,v)] &\mapsto& [(0,v)] \,. }$Notice how, in this coordinatization, the projective spaces are horizontally included by adjoining a 0-coordinate to the *left* of the sequence 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:

$k P^\infty \;\simeq\; k P^{\infty + 1} \;\simeq\; Th \big( \mathcal{L}_{k P^\infty} \big) \,.$See for instance this Lemma at *universal complex orientation on MU*.

Brought in the *dual* tautological line bundle (here).

Tried to clearly sort out the evident subtlety of when the tautological bundle appears, and when its dual. Unless I am mixed up, we have:

The evident map out of the total space of the tautological line bundle exhibits the blow-up.

The evident map out of the total space of the

*dual*tautological line bundle exhibits its Thom space as the next higher projective space.

added cross-link with *Möbius strip*

I have tried my hand at an illustration: here

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