Proposition 10.2.11 (One-point compactification). For every topological space \(X\), fiberwise one-point compactification defines a symmetric monoidal functor \[ S_X^{(-)}\colon \Vect _{\R }(X)^{\simeq } \longrightarrow (\An _{/\Pi _{\infty }(X)})_* \simeq \Fun (\Pi _{\infty }(X),\An _*). \] These functors are the components of a natural transformation \[ S^{(-)}\colon \Vect _{\R }(-)^{\simeq } \Longrightarrow (\An _{/\Pi _{\infty }(-)})_* \] between functors \(\Top \catop \to \CMon (\Cat _{\infty })\), where the source is regarded as taking values in symmetric monoidal \(\infty \)-groupoids. For \(X=\pt \), there is an isomorphism \[ \Vect _{\R }(\pt )^{\simeq }\cong \bigsqcup _{n\geq 0}\bB O(n) \] of commutative monoids, where addition is induced by the block-sum homomorphisms \(O(k)\times O(l)\to O(k+l)\). The restriction of \(S_{\pt }^{(-)}\) to the rank \(n\) component is classified by the standard pointed action of \(O(n)\) on \(S^{\R ^n}\).
Proof of Proposition 10.2.11. Fix a topological space \(X\). Since every morphism in the source of \(\Phi _{X,q}\) is a bundle isomorphism, the cocone of Lemma 10.5.7 factors through the groupoid core of \(\Fun (\Pi _{\infty }(X),\An _*)\). By the defining colimit of Definition 10.2.10, it therefore induces a symmetric monoidal functor \[ S_X^{(-)}\colon \Vect _{\R }(X)^{\simeq } \longrightarrow \Fun (\Pi _{\infty }(X),\An _*) \simeq (\An _{/\Pi _{\infty }(X)})_*. \] In simplicial degree zero, this functor sends a vector bundle \(E\to X\) to the spherical fibration \(\xi _E\). The naturality in \(X\) asserted in the proposition follows from the naturality of the cocones in Lemma 10.5.7.
It remains to identify the rank components and the restriction of \(S_{\pt }^{(-)}\). Real vector bundles over a simplex are trivial, and the automorphisms of the trivial rank \(n\) bundle form the singular simplicial group of \(\GL _n(\R )\). Its realization is \(\bB \GL _n(\R )\). Polar decomposition gives a homotopy equivalence \(O(n)\hookrightarrow \GL _n(\R )\), and hence an isomorphism \[ \bB O(n)\cong \bB \GL _n(\R ). \] Thus the rank \(n\) component of \(\Vect _{\R }(\pt )^{\simeq }\) is \(\bB O(n)\). The construction sends a linear isomorphism to its extension over one-point compactifications, so its restriction to this component classifies the standard pointed action of \(O(n)\) on \(S^{\R ^n}\). The block-sum maps give compatibility with the commutative-monoid structures. □
Generated from the authoritative LaTeX source.