The construction of the previous section is tied to point-set models, does not exhibit the coherence of one-point compactification with respect to direct sums, and treats each bordism spectrum separately. More importantly, it applies only to honest vector bundles. The normal bundle \(\nu \) of an embedding \(M \hookrightarrow \R ^{n+k}\) of a closed \(n\)-manifold is not an invariant of \(M\): enlarging the ambient space changes it by trivial summands. What is intrinsic is the stable normal bundle, the virtual vector bundle \[ \nu - \ul {\R ^k} \quad = \quad \ul {\R ^n} - T_M \] of virtual rank \(0\), which is the class classified by a map \(M \to \bB O\). Closely related is the virtual bundle \(-T_M = \nu - \ul {\R ^{n+k}}\) of virtual rank \(-n\), which is the one occurring in Atiyah duality in Section 11.3. Neither is an honest vector bundle, and neither has a Thom space in the sense of Definition 10.1.2.
The remedy is to describe the Thom construction in a way which never refers to the total space of the bundle. Recall the picture from the introduction: a rank \(n\) vector bundle over \(X\) determines a family of \(n\)-spheres parametrized by \(X\), and the Thom space should be the colimit of that family. Formulated this way, the input is a functor into pointed animae, and once we stabilize it becomes a functor into invertible spectra, where inverses exist and allow for an extension to virtual bundles.
10.2.1 Thom animae
We start by making precise the claim that Thom spaces are colimits of parametrized families of spheres.
Definition 10.2.1 (Spherical fibration). Let \(B\) be an anima and let \(n \geq 0\). A rank \(n\) spherical fibration over \(B\) is a functor \(\xi \colon B \to \An _*\) which lands in the full subcategory spanned by the \(n\)-sphere \(S^n\). A spherical fibration is a rank \(n\) spherical fibration for some \(n\).
Definition 10.2.2 (Thom anima). We define the Thom anima of a spherical fibration \(\xi \colon B \to \An _*\) as its colimit: \[ \Th (\xi ) \quad := \quad \colim (\xi \colon B \to \An _*). \]
Recall from Corollary 23.2.8 the straightening-unstraightening equivalence \(\An _{/B} \simeq \Fun (B,\An )\). Passing to pointed objects, this provides an equivalence \[ \Fun (B,\An _*) \simeq \Fun (B,\An )_* \simeq (\An _{/B})_*, \] so that a rank \(n\) spherical fibration over \(B\) is the same as a pair \((p,s)\) of a morphism \(p\colon A \to B\) of animae and a section \(s\colon B \to A\) of \(p\) such that each fiber \(A_b\) of \(p\) is isomorphic to the \(n\)-sphere \(S^n\).
Example 10.2.3 (Sphere bundle). Let \(X\) be a topological space and consider an \(n\)-dimensional sphere bundle \(Y \to X\) over \(X\), i.e. a fiber bundle whose generic fiber is \(S^n\) and whose structure group is \(\mathrm {Homeo}_*(S^n)\). Then the underlying map of animae \(\Pi _{\infty }(Y) \to \Pi _{\infty }(X)\) is a spherical fibration: by Proposition 2.3.19 and Corollary 2.4.15 the fibers of this map are the spheres \(S^n\).
Example 10.2.4 (Spherical fibration of vector bundle). Given a rank \(n\) vector bundle \(p\colon E \to X\), we may apply the previous example to the fiberwise one-point compactification \(S^E \to X\) from Construction 10.1.1. We denote the resulting spherical fibration by \[ \xi _E\colon \Pi _{\infty }(X) \to \An _*. \] Informally, \(\xi _E\) is the functor which sends a point \(x \in X\) to \(\Pi _{\infty }(S^{E_x})\), where \(S^{E_x}\) is the one-point compactification of the fiber \(E_x\) of \(p\) over \(x\).
We now compare the geometric Thom space with the homotopical Thom anima:
Proposition 10.2.5. Let \(p\colon E\to X\) be a vector bundle over a paracompact Hausdorff space of the homotopy type of a cell complex. Then the underlying pointed anima of its Thom space is isomorphic to the Thom anima of \(\xi _E\): \[ \Pi _{\infty }(\Th (E)) \cong \Th (\xi _E). \]
Proof. We first identify the colimit of any pointed family with the cofiber of its section. We then compare this homotopy cofiber with the point-set quotient defining the classical Thom space.
Set \(B:=\Pi _{\infty }(X)\). Under the straightening-unstraightening equivalence \(\Fun (B,\An _*) \iso (\An _{/B})_*\), the spherical fibration \(\xi _E\) corresponds to the pointed object of \(\An _{/B}\) given by \[ \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \longrightarrow \Pi _{\infty }(X). \] More generally, suppose that a functor \(\xi \colon B\to \An _*\) corresponds to a pointed object \(B\xrightarrow {s}A\to B\) of \(\An _{/B}\). By Proposition 23.5.2, the colimit of the underlying functor \(B\to \An \) is \(A\), while the colimit of the constant basepoint diagram is \(B\). Since colimits in pointed objects are obtained by forming a pushout over the basepoint, it follows that \[ \colim _B\xi \cong \cofib (s\colon B\to A). \] Applied to \(\xi _E\), this gives an isomorphism \[ \Th (\xi _E) \cong \cofib \big ( \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \big ). \]
It remains to identify this cofiber with the classical Thom space. Choose a fiberwise metric on \(E\), which exists by paracompactness. It identifies the fiberwise one-point compactification \(S^E\) with the unit sphere bundle of \(E\oplus \ul {\R }\). Under this identification, the section \(s_{\infty }\) is the section given by the north pole in each fiber. The pair \((S^n,\infty )\) admits an NDR structure (cf. Corollary 2.3.11), which may be chosen \(O(n)\)-equivariantly and therefore induces an NDR structure on \((S^E,s_{\infty }(X))\). Moreover, \(S^E\) has the homotopy type of a cell complex, since it is the total space of a fibration over a space of the homotopy type of a cell complex with fiber \(S^n\); see Milnor (1959), Theorem 3. Thus \(X\) and \(S^E\) satisfy the hypotheses of Corollary 2.3.11, giving the isomorphism \[ \cofib \big ( \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \big ) \simeq \Pi _{\infty }\big (S^E/s_{\infty }(X)\big ) = \Pi _{\infty }(\Th (E)), \] as desired. □
10.2.2 Thom spectra
The homotopical perspective of Thom spaces directly generalizes to the stable setting.
Definition 10.2.6 (Stable spherical fibration). Let \(\Pic (\S ) \subseteq \Sp ^{\simeq }\) denote the full subcategory spanned by shifts \(\S [n]\) of the sphere spectrum. For an anima \(A\), we define a stable spherical fibration on \(A\) to be a functor \(\xi \colon A \to \Sp \) which lands in \(\Pic (\S )\). For an integer \(n \in \Z \) we say that \(\xi \) has rank \(n\) if it in fact lands in the full subanima of \(\Pic (\S )\) spanned by \(\S [n]\).
One can show that objects of \(\Pic (\S )\) are precisely the invertible spectra: those spectra \(X\) for which there is a spectrum \(X^{-1}\) and an isomorphism \(X \otimes X^{-1} \cong \S \); see for example [Hovey et al. (1997)]. In particular, \(\Pic (\S )\) is precisely the Picard anima of \(\Sp \) in the sense of Definition 11.1.4 below.
If \(\xi \colon A \to \An _*\) is a rank \(n\) spherical fibration, then \(\Sigma ^{\infty }\xi \) is a rank \(n\) stable spherical fibration. In particular, a rank \(n\) vector bundle \(E\to X\) determines the stable spherical fibration \(\Sigma ^{\infty }\xi _E\) on \(\Pi _{\infty }(X)\).
Definition 10.2.7 (Thom spectrum). Given a stable spherical fibration \(\xi \colon A \to \Sp \), we define its Thom spectrum as its colimit: \[ \th (\xi ) \quad := \quad \colim (\xi \colon A \to \Sp ). \]
For a vector bundle \(E \to X\), we write \[ \th (E) := \Sigma ^\infty \Th (\xi _E) \] for its Thom spectrum. Since the suspension spectrum functor \(\Sigma ^{\infty }(-)\colon \An _* \to \Sp \) preserves colimits, this agrees with \(\th (\Sigma ^{\infty }\xi _E)\), and by Proposition 10.2.5 it is the suspension spectrum of the underlying pointed anima of the classical Thom space. Throughout, \(\Th \) denotes a Thom space or Thom anima and \(\th \) a Thom spectrum.
Construction 10.2.8 (Multiplicative structures on Thom spectra). Let \(A\) be a small commutative monoid in animae, regarded as a symmetric monoidal \(\infty \)-category, and let \[ \xi \colon A\longrightarrow \Pic (\S )\subseteq \Sp \] be a symmetric monoidal functor. We equip the Thom spectrum \(\th (\xi )\) with a canonical commutative ring spectrum structure, following Antolín-Camarena and Barthel (2019).
By Corollary 16.2.8, the functor category \(\Fun (A,\Sp )\) admits the Day convolution symmetric monoidal structure. The universal property of Day convolution from Lemma 16.1.4 identifies the underlying lax symmetric monoidal functor of \(\xi \) with a commutative algebra in \(\Fun (A,\Sp )\). The colimit functor \[ \colim _A\colon \Fun (A,\Sp )\longrightarrow \Sp \] is pointwise left Kan extension along the unique symmetric monoidal functor \(A\to *\), so it is symmetric monoidal by Lemma 16.2.11. Here \(\Fun (*,\Sp )\) is identified with \(\Sp \) equipped with its usual tensor product. It therefore carries \(\xi \) to a commutative algebra in spectra whose underlying spectrum is \[ \colim _A\xi =\th (\xi ). \]
Remark 10.2.9. There is an analogous multiplicative Thom-spectrum construction with \(\Comm \) replaced by another \(\infty \)-operad, once the corresponding operadic Day convolution has been developed. We do not need this generalization here. For the case of the little-cubes operads \(\Ee _n\), see Antolín-Camarena and Barthel (2019).
The symmetric monoidal stable spherical fibrations used below arise from the stable J-homomorphism. We apply the construction to bordism spectra in Construction 10.2.25.
10.2.3 The J-homomorphism
As previously indicated, passing to spectra allows us to extend the definition of Thom spectra to virtual vector bundles. The idea is simple: the assignment \(E\mapsto \Sigma ^{\infty }\xi _E\) is symmetric monoidal and lands in invertible spectra, so it should extend over group completion. The subtlety is coherence: the familiar homeomorphisms \(S^{V\oplus W}\cong S^V\wedge S^W\) must assemble into a symmetric monoidal functor from vector spaces to pointed animae.
For a topological space \(X\), let \(\Vect _{\R }^{\disc }(X)\) denote the 1-category of finite-rank real vector bundles over \(X\) and morphisms of vector bundles. By Corollary 19.7.10, direct sum and pullback give a simplicial diagram \[ [q]\longmapsto \Vect _{\R }^{\disc }(X\times \abs {\Delta ^q})^{\simeq } \qin \CMon (\An ). \]
Definition 10.2.10. We define the moduli anima of real vector bundles over \(X\) by \[ \Vect _{\R }(X)^{\simeq } := \colim _{[q]\in \simp \catop } \Vect _{\R }^{\disc }(X\times \abs {\Delta ^q})^{\simeq } \qin \CMon (\An ). \] This colimit is computed on underlying animae and is functorial in \(X\), giving a functor \(\Vect _{\R }(-)^{\simeq }\colon \Top \catop \to \CMon (\An )\). When \(X=\pt \), we refer to \(\Vect _{\R }(\pt )^{\simeq }\) as the moduli anima of finite-dimensional real vector spaces.
The following proposition supplies all the coherence needed below. Its proof occupies Section 10.5.
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 \(S^{\R ^n}\).
We also write \(S^{(-)}\) for the component \(S_{\pt }^{(-)}\) when the base is a point.
Definition 10.2.12. Write \(O:=\colim _n O(n)\) for the stable orthogonal group and \(\bB O\) for its classifying anima. We define \[ \BOP := \big (\Vect _{\R }(\pt )^{\simeq }\big )^{\grp } \qin \CGrp (\An ) \] to be the group completion of the commutative monoid of finite-dimensional real vector spaces. The telescope calculation of Corollary 9.4.3, with real vector spaces in place of complex ones, identifies its underlying anima as \[ \BOP \simeq \Z \times \bB O. \] The only additional point in applying Theorem 5.5.2 is that \(\pi _1(\bB O) \cong \pi _0(O) \cong C_2\) is abelian. The virtual-rank-zero component has the presentation \[ \bB O\cong \colim _n\bB O(n), \] where the transition maps add a trivial line. Under this presentation, the map from the \(n\)-th stage to the virtual-rank-zero component sends \(V\) to \(V-\R ^n\).
Construction 10.2.13 (Stable J-homomorphism). Applying reduced suspension spectra to Proposition 10.2.11 gives a symmetric monoidal functor \[ \Vect _{\R }(\pt )^{\simeq } \xrightarrow {\,S^{(-)}\,} \An _* \xrightarrow {\Sigma ^\infty } \Sp . \] Its image lies in \(\Pic (\S )\), since \(\Sigma ^\infty S^V\cong \S [\dim (V)]\) is invertible. Since tensor product of spectra turns \(\Pic (\S )\) into a commutative group in animae, the resulting functor to \(\Pic (\S )\) extends uniquely over group completion, giving a symmetric monoidal functor \[ J\colon \BOP \longrightarrow \Pic (\S ). \] We call \(J\) the stable J-homomorphism. We use the same notation for its restriction to any component of \(\BOP \), in particular for \[ J\colon \bB O \simeq \{0\}\times \bB O \longrightarrow \Pic (\S ). \]
Remark 10.2.14 (Relation to the classical J-homomorphism). The connected component of \(\S \) in \(\Pic (\S )\) is equivalent to \(\bB \GL _1(\S )\), where \(\GL _1(\S ) := \Aut _{\Sp }(\S )\) is the automorphism group of the sphere spectrum. The restriction of \(J\) to the virtual-rank-zero component lands in this connected component. Passing to loops gives a group homomorphism \(O\to \GL _1(\S )\) in \(\An \), whose effect on positive homotopy groups is the classical stable J-homomorphism \(\pi _n(O)\to \pi _n(\S )\). We refer to Ravenel (1986), Chapter 1 for a classical discussion of this map.
The complex analogue is the composite \(\BUP \to \BOP \xrightarrow {J}\Pic (\S )\) induced by forgetting the complex structure.
The point of constructing \(J\) on the full group completion is that it assigns stable spherical fibrations, and hence Thom spectra, to virtual vector bundles of arbitrary rank.
Definition 10.2.15 (Virtual vector bundle). Let \(A\) be an anima. A virtual vector bundle on \(A\) is a map of animae \[ V\colon A \longrightarrow \BOP . \] Its rank is the composite \(\rk (V)\colon A \to \Z \) with the projection to \(\Z \), and we say that \(V\) has rank \(d \in \Z \) if this composite is constant with value \(d\). A virtual vector bundle on a topological space \(X\) means one on \(\Pi _{\infty }(X)\).
Since \(\BOP \) is a commutative group in animae, so is the anima \(\Hom _{\An }(A,\BOP )\) of virtual vector bundles on \(A\). We write \(V + W\) for the sum and \(-V\) for the inverse; ranks add, and \(\rk (-V) = -\rk (V)\).
Construction 10.2.16 (Thom spectrum of a virtual vector bundle). Let \(V\) be a virtual vector bundle on an anima \(A\). Composing with the J-homomorphism gives a stable spherical fibration \[ \xi _V \quad := \quad J \circ V\colon A \longrightarrow \Pic (\S ). \] If \(V\) has rank \(d\), then so does \(\xi _V\), since \(J\) carries the class of \(\R \) to \(\S [1]\). We define the Thom spectrum of \(V\) as \[ \th (V) \quad := \quad \th (\xi _V) \quad = \quad \colim \big (J \circ V\colon A \to \Sp \big ). \]
Remark 10.2.17. As \(J\) is a homomorphism of commutative groups in animae, the assignment \(V \mapsto \xi _V\) carries sums to pointwise tensor products and inverses to pointwise inverses: \[ \xi _{V+W} \cong \xi _V \otimes \xi _W \qquad \text {and}\qquad \xi _{-V} \cong \xi _V^{-1}. \] The Thom spectrum itself is of course not additive: \(\th (V+W)\) is a colimit over \(A\), not a tensor product of colimits.
We next relate this construction to honest vector bundles. A pointed action of a topological group \(G\) on a topological space \(Y\), together with a principal \(G\)-bundle \(P\to X\), gives an associated pointed bundle \(P\times _GY\to X\). Passing to fundamental animae and applying pointed straightening produces a functor \(\Pi _{\infty }(X)\to \An _*\). For a universal principal bundle, we refer to the resulting functor \(\bB G\to \An _*\) as the functor classified by the pointed \(G\)-action on \(Y\).
Construction 10.2.18 (One-point compactification of vector spaces). By Proposition 10.2.11, the rank \(n\) component of \(\Vect _{\R }(\pt )^{\simeq }\) is isomorphic to \(\bB O(n)\). We denote the restriction of one-point compactification to this component by \[ J_n^{\mathrm {un}}\colon \bB O(n) \longrightarrow \An _*. \] It is the functor classified by the standard pointed action of \(O(n)\) on \(S^{\R ^n}\). Consequently, the restriction of \(J\) along the rank \(n\) component \(\bB O(n)\to \BOP \) is \(\Sigma ^\infty J_n^{\mathrm {un}}\). Its restriction along the stable map \[ \bB O(n)\longrightarrow \{0\}\times \bB O\subseteq \BOP , \qquad V\longmapsto V-\R ^n, \] is \(\Sigma ^\infty J_n^{\mathrm {un}}[-n]\).
Lemma 10.2.19. Let \(X\) be a paracompact Hausdorff space and let \(f\colon X \to \Gr _n(\R ^{\infty })\) be a continuous map. Let \(p\colon E \to X\) be the rank \(n\) vector bundle classified by \(f\). Then the map \(\xi _E\colon \Pi _{\infty }(X) \to \An _*\) is given by the composite \[ \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(f)} \Pi _{\infty }(\Gr _n(\R ^{\infty })) \iso \bB O(n) \xrightarrow {J_n^{\mathrm {un}}} \An _*, \] where the middle isomorphism is the one from Proposition 9.2.11.
Proof. By definition, \(p\colon E \to X\) is the pullback of the universal bundle \(p_{\univ }\colon E_{\univ } \to \Gr _n(\R ^{\infty })\). The induced square of sphere bundles is carried to a pullback of animae by Proposition 2.3.19. Under straightening-unstraightening, pullback corresponds to precomposition, so it suffices to prove the claim for \(p_{\univ }\). The Stiefel manifold \(V_n(\R ^\infty )\) of orthonormal \(n\)-frames is contractible, and its projection to \(\Gr _n(\R ^\infty )\) is a principal \(O(n)\)-bundle. It therefore identifies \(\Pi _{\infty }(\Gr _n(\R ^\infty ))\) with \(\bB O(n)\) by Proposition 9.2.11. Under this identification, the fiberwise one-point compactification of the universal bundle is the associated pointed sphere bundle \[ V_n(\R ^\infty ) \times _{O(n)} S^n \longrightarrow \Gr _n(\R ^\infty ). \] Its straightening is precisely \(J_n^{\mathrm {un}}\). □
Example 10.2.20 (The class of a vector bundle). Let \(X\) be a paracompact Hausdorff space and let \(E \to X\) be a rank \(n\) vector bundle, classified by a map \(f\colon X \to \Gr _n(\R ^{\infty })\). Combining the identification \(\Pi _{\infty }(\Gr _n(\R ^{\infty })) \iso \bB O(n)\) of Proposition 9.2.11 with the preceding construction, we obtain a virtual vector bundle \[ [E]\colon \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(f)} \bB O(n) \subseteq \Vect _{\R }(\pt )^{\simeq } \longrightarrow \BOP \] of rank \(n\). For the trivial bundle \(\ul {\R ^N}\) this is the constant map with value \(\R ^N\).
This construction is compatible with direct sums. Indeed, under the identifications \(\Pi _{\infty }(\Gr _n(\R ^{\infty }))\cong \bB O(n)\), the Whitney-sum map corresponds to the map induced by the block-sum homomorphism \(O(n)\times O(m)\to O(n+m)\). To see this, observe that the orthonormal frame bundle of \(E\oplus E'\) is obtained from the product over \(X\) of the orthonormal frame bundles of \(E\) and \(E'\) by extension of structure group along this homomorphism. Since the addition on \(\Vect _{\R }(\pt )^{\simeq }\) is induced by block sum, it follows that \[ [E\oplus E']\cong [E]+[E']. \]
Finally, we record the compatibility with the construction for honest vector bundles, and the interaction between adding trivial summands and shifts; we will use these properties in our proof of Atiyah duality in Section 11.3.
Lemma 10.2.21. Let \(X\) be a paracompact Hausdorff space of the homotopy type of a cell complex and let \(E \to X\) be a rank \(n\) vector bundle. Then there is an isomorphism \(\xi _{[E]} \cong \Sigma ^{\infty }(\xi _E)\) of stable spherical fibrations, and hence an isomorphism \(\th ([E]) \cong \th (E)\). More generally, for every \(N \geq 0\) we have \(\xi _{[E]-[\ul {\R ^N}]} \cong \Sigma ^{\infty }(\xi _E)[-N]\) and therefore \[ \th \big ([E]-[\ul {\R ^N}]\big ) \quad \cong \quad \th (E)[-N] \quad \cong \quad \Sigma ^{\infty -N}\Th (E). \]
Proof. By Construction 10.2.18, the restriction of \(J\) to the rank \(n\) component is \(\Sigma ^{\infty }J_n^{\mathrm {un}}\). The first isomorphism is therefore precisely Lemma 10.2.19, and passing to colimits gives \(\th ([E]) \cong \Sigma ^{\infty }\Th (\xi _E) = \th (E)\), since \(\Sigma ^{\infty }\) preserves colimits.
The virtual vector bundle \([\ul {\R ^N}]\) is constant with value \(\R ^N\), so \(\xi _{[\ul {\R ^N}]}\) is constant with value \(\Sigma ^{\infty }S^{\R ^N} \cong \S [N]\). By Remark 10.2.17 we thus obtain \[ \xi _{[E]-[\ul {\R ^N}]} \cong \Sigma ^{\infty }(\xi _E) \otimes \S [-N] \cong \Sigma ^{\infty }(\xi _E)[-N], \] and the remaining claims follow because shifts commute with colimits. □
10.2.4 Bordism spectra
We now apply the Thom-spectrum construction to bordism spectra. We start with \(\MO \), which comes from the virtual-rank-zero component of \(\BOP \), and relate it to the prespectrum \(\MO ^{\pre }\) constructed in Section 10.1.
Definition 10.2.22 (Unoriented bordism spectrum). We define the unoriented bordism spectrum \(\MO \) as the Thom spectrum of the restriction of \(J\) to the virtual-rank-zero component: \[ \MO \quad := \quad \th (J\vert _{\bB O}) \quad = \quad \colim \big (J\colon \bB O\to \Sp \big ). \]
Theorem 10.2.23. The spectrum associated to the prespectrum \(\MO ^{\pre }\) of Construction 10.1.10 via Proposition 4.3.29 is \(\MO \): \[ (\MO ^{\pre })^{\mathrm {sp}} \cong \MO . \]
Proof. We first identify the levels and structure maps of the classical prespectrum with the unstable Thom construction. We then pass to the colimit presentation of the virtual-rank-zero component of \(\BOP \).
Step 1: Consider the functor \(\xi _{E_n} \colon \Pi _{\infty }(\Gr _n(\R ^{\infty })) \to \An _*\) associated to \(\gamma _n\). Applying Lemma 10.2.19 to the identity of \(\Gr _n(\R ^\infty )\) identifies \(\xi _{E_n}\) with \(J_n^{\mathrm {un}}\). In particular, Proposition 10.2.5 provides an isomorphism \[ \MO ^{\pre }_n = \Pi _{\infty }(\Th (\gamma _n)) \cong \Th (J_n^{\mathrm {un}}) \] of animae. The identification \(E_{n+1}\vert _{\Gr _n(\R ^{\infty })} \cong E_n \times \R \) corresponds to the monoidal identification \[ J_{n+1}^{\mathrm {un}}\vert _{\bB O(n)} \cong \Sigma J_n^{\mathrm {un}}. \] Thus the structure maps of \(\MO ^{\pre }\) correspond to the canonical comparison maps \[ \Sigma \Th (J_n^{\mathrm {un}}) \cong \Th \big (J_{n+1}^{\mathrm {un}}\vert _{\bB O(n)}\big ) \longrightarrow \Th (J_{n+1}^{\mathrm {un}}). \]
Step 2: Applying Theorem 21.2.11 to the presentation \(\bB O \simeq \colim _{n \in \N } \bB O(n)\) of Definition 10.2.12 gives \[ \MO \cong \colim _n \Sigma ^{\infty }\Th (J_n^{\mathrm {un}})[-n]. \] The structure maps in this colimit diagram are precisely those of \(\MO ^{\pre }\), so the right-hand side is \((\MO ^{\pre })^{\mathrm {sp}}\). □
The abstract definition also makes the variants of \(\MO \) formal.
Definition 10.2.24 (The Thom spectrum \(MB\)). Let \(\phi \colon B \to \BOP \) be a map of animae. We define its associated Thom spectrum \(MB\) by \[ MB := \colim \big (B\xrightarrow {\phi }\BOP \xrightarrow {J}\Sp \big ). \] Equivalently, \(\phi \) is a virtual vector bundle on \(B\), and \(MB=\th (\phi )\) in the notation of Construction 10.2.16. The notation suppresses \(\phi \), which will always be clear from the context. We allow \(B\) to be an arbitrary anima. For the Pontryagin–Thom theorem of Section 10.3, we will specialize to maps which land in the virtual-rank-zero component \(\bB O\subseteq \BOP \).
Construction 10.2.25 (Multiplicative structures on bordism spectra). Let \(B\) be a commutative monoid in animae and let \[ \phi \colon B\longrightarrow \BOP \] be a morphism of commutative monoids. Since the J-homomorphism is symmetric monoidal, so is the composite \[ B\xrightarrow {\phi }\BOP \xrightarrow {J}\Pic (\S )\subseteq \Sp . \] Consequently, Construction 10.2.8 equips \(MB\) with a canonical commutative ring spectrum structure.
Example 10.2.26. A continuous group homomorphism \(G \to O\) induces a map \(\bB G \to \bB O\), and we write \(MG\) for the resulting Thom spectrum. Taking \(B = \bB O\) with \(\phi \) the identity recovers \(\MO \). Further examples are \[ \MSO , \quad \MU , \quad \mathrm {MSU}, \quad \mathrm {MSp}, \quad \text {and}\quad \mathrm {MSpin}. \] Here \(\MU \) is the complex bordism spectrum. For \(\MO \) and \(\MU \), the defining maps to \(\BOP \) are restrictions of morphisms of commutative groups constructed above. Hence Construction 10.2.25 equips these Thom spectra with commutative ring structures. The other classical Thom spectra in the display also admit commutative ring structures, but their construction requires compatible infinite-loop refinements of the maps \(\bB G\to \bB O\), which we do not develop here.
The full group completions give the periodic unoriented and complex bordism spectra \[ \MOP :=\th \big (J\colon \BOP \to \Sp \big ) \qquadtext {and}\qquad \MUP :=\th \big (\BUP \longrightarrow \BOP \xrightarrow {J}\Sp \big ), \] respectively. Thus \(\MOP \) is obtained from the identity of \(\BOP \), while \(\MUP \) is obtained from the forgetful map \(\BUP \to \BOP \). Restricting these defining diagrams to their virtual-rank-zero components recovers \(\MO \) and \(\MU \). Translation by the classes of \(\R \) and \(\C \) permutes the components of \(\BOP \) and \(\BUP \) while shifting the associated stable spherical fibrations by \(1\) and \(2\), respectively. It follows that there are isomorphisms of spectra \[ \MOP \cong \MOP [1] \qquadtext {and}\qquad \MUP \cong \MUP [2], \] which explains the terminology.
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