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added cross-link with Möbius strip
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.
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.
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).
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…
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