This section interprets orientations of virtual vector bundles as trivializations of local systems of lines, and derives the resulting Thom isomorphism. Throughout, we fix a commutative ring spectrum \(R \in \CAlg (\Sp )\).
A virtual vector bundle \(V\) of rank \(d\) on an anima \(A\) determines a stable spherical fibration \(\xi _V\colon A \to \Pic (\S )\). After shifting by \(-d\) and tensoring with \(R\), its fibers are \(R\)-modules isomorphic to \(R\). An orientation is a coherent choice of such isomorphisms. This formulation applies directly to virtual bundles and specializes to the classical notion of a Thom class for an honest bundle. It follows the orientation theory of Ando et al. (2014); for the classical account, see May (1999), Chapter 23, Section 5.
Definition 10.4.1 (\(R\)-lines). We denote by \[ \Line _R \subseteq \Mod _R^{\simeq } \] the full subanima spanned by those \(R\)-modules that are isomorphic to \(R\). Its objects are called \(R\)-lines.
Remark 10.4.2. The pointed connected anima \(\Line _R\) is the classifying anima of the group of \(R\)-linear automorphisms of \(R\): \[ \Line _R \simeq \bB \Aut _R(R). \]
Example 10.4.3. For an ordinary commutative ring \(k\), the group \(\Aut _{Hk}(Hk)\) is the discrete group \(k^\times \). In particular, \[ \Line _{H\F _2} \simeq * \qquadtext {and} \Line _{H\Z } \simeq \bB C_2. \]
Definition 10.4.4 (Local systems and trivializations). A local system of \(R\)-lines on an anima \(A\) is a functor \(\xi \colon A \to \Line _R\). A trivialization of \(\xi \) is an isomorphism \(\xi \cong R_A\), where \(R_A\) is the constant functor with value \(R\).
Definition 10.4.5 (Orientation). Let \(V\) be a virtual vector bundle of rank \(d\) on an anima \(A\). Its associated local system of \(R\)-lines is \[ \xi _{V,R}\colon A \xrightarrow {\;\xi _V[-d]\;} \Line _{\S } \xrightarrow {-\otimes R} \Line _R. \] Here \(\xi _V[-d]\) lands in the virtual-rank-zero component \(\Line _{\S }\subseteq \Pic (\S )\). An \(R\)-orientation of \(V\) is a trivialization \(\xi _{V,R} \cong R_A\).
An \(R\)-orientation of \(V\) induces a morphism of \(R\)-modules \[ \th (V)[-d]\otimes R \cong \colim _A\xi _{V,R} \cong R[A] \longrightarrow R, \] where the last map is the augmentation. By adjunction, this corresponds to a class \[ u_V\colon \th (V)\longrightarrow R[d], \] which we call the Thom class of the oriented virtual bundle \(V\).
Lemma 10.4.6 (Orientations of inverse virtual bundles). Let \(V\) be a virtual vector bundle on \(A\). An \(R\)-orientation of \(V\) canonically induces an \(R\)-orientation of \(-V\).
Proof. By Remark 10.2.17, including the normalization by virtual rank, there is an isomorphism \(\xi _{-V,R} \cong \xi _{V,R}^{-1}\). Inverting a trivialization of \(\xi _{V,R}\) gives the required trivialization. □
For an honest vector bundle, this recovers the familiar definition in terms of a Thom class.
Proposition 10.4.7. Let \(E \to X\) be a rank \(n\) vector bundle over a paracompact Hausdorff space of the homotopy type of a cell complex. Under the identification \(\th ([E])\cong \Sigma ^\infty \Th (E)\) of Lemma 10.2.21, an \(R\)-orientation of \([E]\) is the same as a class \(\theta \in \widetilde R^n(\Th (E))\) whose restriction to every fiber is a unit in \(\pi _0(R)\).
Proof. A class \(\theta \in \widetilde R^n(\Th (E))\) is a morphism \(\th ([E])[-n]\otimes R\to R\) of \(R\)-modules. Since the source is the colimit of \(\xi _{[E],R}\), adjunction identifies this with a morphism \(\xi _{[E],R}\to R_{\Pi _\infty (X)}\) of local systems. It is an isomorphism precisely when each restriction \(\theta _x\in \pi _0(R)\) is a unit. □
Example 10.4.8. Since \(\Line _{H\F _2}\) is contractible, every virtual vector bundle of fixed rank has a unique \(H\F _2\)-orientation. For \(R=H\Z \) and an honest real vector bundle \(E\to X\), the local system \(\xi _{[E],H\Z }\) is classified by \[ X \longrightarrow \bB O(n) \xrightarrow {\det } \bB C_2. \] Thus an \(H\Z \)-orientation is a classical orientation, equivalently a nullhomotopy of \(w_1(E)\).
Example 10.4.9 (The universal \(\MO \)-orientation). The universal rank-zero virtual bundle on \(\bB O\) is canonically \(\MO \)-oriented. Indeed, for every \(W\in \bB O\), symmetric monoidality of \(J\) and the definition \(\MO =\colim _{V\in \bB O}J(V)\) give \[ J(W)\otimes \MO \cong \colim _{V\in \bB O}J(W+V) \cong \MO . \] The second isomorphism is induced by the equivalence \(W+(-)\colon \bB O\to \bB O\), and these isomorphisms vary coherently with \(W\). They therefore trivialize the associated local system of \(\MO \)-lines. Pulling this orientation back along the classifying map of an honest real vector bundle shows that every real vector bundle is canonically \(\MO \)-orientable.
Theorem 10.4.10 (Thom isomorphism). Let \(V\) be an \(R\)-oriented virtual vector bundle of rank \(d\) on an anima \(A\). Then there is an isomorphism of \(R\)-modules \[ \th (V)\otimes R \cong (\S [A]\otimes R)[d], \] and an isomorphism of spectra \[ \hom (\th (V),R)\cong \hom (\S [A][d],R). \]
Proof. Passing to colimits in the chosen trivialization \(\xi _{V,R}\cong R_A\) gives \[ \th (V)[-d]\otimes R \cong \colim _A\xi _{V,R} \cong \colim _A R_A \cong R[A], \] which proves the first isomorphism. The second follows by applying \(\hom _R(-,R)\) and using the free-forgetful adjunction between spectra and \(R\)-modules. □
Corollary 10.4.11 (Classical Thom isomorphism). Let \(E\to X\) be a rank \(n\) vector bundle over a paracompact Hausdorff space of the homotopy type of a cell complex, equipped with an \(R\)-Thom class \(\theta \). Then \[ \widetilde R_m(\Th (E))\cong R_{m-n}(X) \qquadtext {and} \widetilde R^m(\Th (E))\cong R^{m-n}(X). \] The inverse of the cohomological isomorphism is the Thom homomorphism \(x\mapsto x\smile \theta \).
Proof. Apply Theorem 10.4.10 to \(V=[E]\) and use Proposition 10.4.7, Lemma 10.2.21. The description of the cohomological map follows from the Thom diagonal \[ \Delta _E\colon \Th (E)\longrightarrow X_+\wedge \Th (E), \qquad [v]\longmapsto \big (p(v),[v]\big ). \] □
Remark 10.4.12 (The mod \(2\) homology of \(\MO \)). The universal rank-zero virtual bundle on \(\bB O\) has a unique \(H\F _2\)-orientation. The Thom isomorphism gives \[ H\F _2 \otimes \MO \; \cong \; H\F _2[\bB O] \] of \(H\F _2\)-modules. This computes the mod \(2\) homology of \(\MO \) in terms of that of \(\bB O\), whose mod \(2\) cohomology is the polynomial algebra \(\F _2[w_1,w_2,\ldots ]\) on the Stiefel–Whitney classes. It is the first step of Thom’s calculation of \(\pi _*(\MO )\) recorded in Remark 10.3.8, but not the whole of it: splitting \(\MO \) into a wedge of shifts of \(H\F _2\), and identifying the multiplicative structure of the coefficient ring, requires further input.
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