Complex oriented cohomology theoriesOrientations

3 Orientations

Definition 3.1

Let EE be a ring spectrum, and ξ:VX\xi : V \to X a vector bundle. Then VV is EE-oriented if there is a “Thom class” u:Th(ξ)Eu: \mathrm{Th}(\xi ) \to E such that the induced map of EE-modules

\begin{useimager} 
    \[
      \begin{tikzcd}
        E \wedge \Th(\xi) \ar[r, "E \wedge \Delta"] & E \wedge \Th(\xi) \wedge X_+ \ar[r, "E \wedge u \wedge X_+"] & E \wedge E \wedge X_+ \ar[r, "\mu \wedge X_+"] & E \wedge X_+
      \end{tikzcd}
    \]
  \end{useimager}

is an isomorphism.

This induces isomorphisms

E(X+)E(Th(ξ)),E(X+)E(Th(ξ)) E_*(X_+) \cong E^*(\mathrm{Th}(\xi )),\quad E^*(X_+) \cong E^*(\mathrm{Th}(\xi ))

induced by cupping and capping with uu (using the Thom diagonal Δ\Delta ).

Lemma 3.2

Let f:EFf: E \to F be a map of ring spectra. Then an EE-orientation of ξ\xi gives rise to an FF-orientation of ξ\xi by composition.

Proof
[Proof sketch] The key input here is that an EE-module map φ:EAEB\varphi : E \wedge A \to E \wedge B functorially induces an FF-module map FAFBF \wedge A \to F \wedge B via the composite

\begin{useimager} 
    \[
      \begin{tikzcd}
        F \wedge A  \ar[r, "F \wedge \iota \wedge A"] & F \wedge E \wedge A \ar[r, "F \wedge \varphi"] & F \wedge E \wedge B \ar[r, "F \wedge f \wedge B"] & F \wedge F \wedge B \ar[r, "\mu_F \wedge B"] & F \wedge B.
      \end{tikzcd}
    \]
  \end{useimager}

Crucially, this construction does not make use of any coherence; if we had some sort of coherence, we could simply apply FE()F \wedge _E (-). One checks that the Thom isomorphism for FF is induced from that for EE via this procedure. Functoriality then ensures the resulting map is also an equivalence.

Proof

Definition 3.3

A cohomology theory EE is complex oriented if there is a choice of a Thom class uξu_\xi for every complex vector bundle ξ:VX\xi : V \to X that

  1. is functorial under pullbacks, i.e. fuξ=ufξf^* u_\xi = u_{f^* \xi }; and

  2. sends direct sums to products, i.e. uξζ=uξuζu_{\xi \boxplus \zeta } = u_\xi \smile u_\zeta .

Example 3.4

MUMU is complex oriented. Indeed, if ξ:VX\xi : V \to X is a complex vector bundle, it is classified by a map f:XBUf: X \to BU, and ξ\xi is the pullback of the universal bundle. Applying Th\mathrm{Th} gives a map

uξ=Th(f):Th(ξ)MU. u_\xi = \mathrm{Th}(f): \mathrm{Th}(\xi ) \to MU.

We claim this is a Thom class. We have to show that the composite

\begin{useimager} 
    \[
      \begin{tikzcd}
        MU \wedge \Th(\xi) \ar[r, "MU \wedge \Delta"] & MU \wedge \Th(\xi) \wedge X_+ \ar[r, "MU \wedge u \wedge X_+"] & MU \wedge MU \wedge X_+ \ar[r, "\mu \wedge X_+"] & MU \wedge X_+
      \end{tikzcd}
    \]
  \end{useimager}

is an isomorphism. This map is obtained by applying Th\mathrm{Th} to

\begin{useimager} 
    \[
      \begin{tikzcd}
        \gamma \oplus \xi \ar[d] \ar[r] & \gamma \oplus \xi \oplus 0 \ar[d] \ar[r] & \gamma \oplus \gamma \oplus 0 \ar[d] \ar[r] & \gamma \oplus 0 \ar[d]\\
        BU \times X \ar[r, "BU \times \Delta"] & BU \times X \times X \ar[r, "BU \times f \times X"] & BU \times BU \times X \ar[r, "\oplus \times X"] & BU \times X
      \end{tikzcd}
    \]
  \end{useimager}

Here γ\gamma is the tautological bundle, and when we write abca \oplus b \oplus c, we really mean π1aπ2bπ3c\pi _1^* a \oplus \pi _2^* b \oplus \pi _3^* c.

We can describe the bottom map as sending (v,x)(v, x) to (v+f(x),x)(v + f(x), x). This has an inverse given by (v,x)(vf(x),x)(v, x) \mapsto (v - f(x), x), so is an isomorphism. Hence the induced map on Thom spaces is also an isomorphism.

The requirement that uξζ=uξuζu_{\xi \boxplus \zeta } = u_\xi \smile u_\zeta comes from the fact that the multiplication map MU×MUMUMU \times MU \to MU is induced by :BU×BUBU\oplus : BU \times BU \to BU.

Lemma 3.5

Let EE be a ring spectrum. Then there is a bijection between

  1. ring maps MUEMU \to E; and

  2. complex orientations of EE.

Proof
Since MUMU is complex oriented, a ring map MUEMU \to E gives a complex orientation of EE. Conversely, if EE is complex oriented, then since MUMU is the Thom spectrum of a complex vector bundle, we get a Thom class u:MUEu: MU \to E. This is a ring map since the product MUMUMUMU \wedge MU \to MU is induced by the direct sum, and the requirement that the Thom class sends direct sums to products is exactly the statement that MUEMU \to E is a ring map.
Proof

Lemma 3.6

Let EE be complex oriented. Then there is a canonical isomorphism E(CP+n)E[u]/un+1E^*(\mathbb {CP}^n_+) \cong E^*[u]/u^{n + 1} and E(CP+)E[ ⁣[u] ⁣]E^*(\mathbb {CP}^\infty _+) \cong E^*[\! [u]\! ]. Note that the choice of uu depends on the complex orientation of EE.

Proof
We first construct the class uu. For n>0n > 0, we know that CPn\mathbb {CP}^n is the Thom space of the tautological bundle over CPn1\mathbb {CP}^{n - 1}. So there is a preferred class uE2(CPn)u \in E^2(\mathbb {CP}^n). For n=1n = 1, the Thom isomorphism theorem tells us this generates E(CP1)E2(S0)E^*(\mathbb {CP}^1) \cong E^{* - 2}(S^0). This proves the lemma for n=1n = 1. Note that inductively applying the theorem proves that E(CP+n)E[u]/un+1E^*(\mathbb {CP}^n_+) \cong E^*[u]/u^{n + 1} as groups, but we want the multiplicative structure as well.

For n>1n > 1, consider the Atiyah–Hirzebruch spectral sequence for E(CPn)E^*(\mathbb {CP}^n). We claim that uu is represented by a generator of H2(CPn;E0)E0H^2(\mathbb {CP}^n; E^0) \cong E^0. Indeed, if it weren't, then its image when pulled back along CP1CPn\mathbb {CP}^1 \hookrightarrow \mathbb {CP}^n would also not be a generator, which contradicts our previous observation.

Now the E2E^2 page of the Atiyah–Hirzebruch spectral sequence for E(CP+n)E^*(\mathbb {CP}^n_+) is generated as a ring by uu and EE^*, both of which are permanent. So the Atiyah–Hirzebruch spectral sequence degenerates and gives the desired isomorphism.

The n=n = \infty case follows from taking the limit.

Proof

Theorem 3.7

There is a bijection between

  1. complex orientations of EE; and

  2. classes uE2(CP)u \in E^2(\mathbb {CP}^\infty ) that restrict to an EE^*-module generator of E2(CP1)E0E^2(\mathbb {CP}^1) \cong E^0.

Proof
[Proof sketch] Our previous computation showed that a complex orientation of EE induces such a class. The other direction is more roundabout. As in our previous argument, the class uu forces the Atiyah–Hirzebruch spectral sequence for E(CP+)E^*(\mathbb {CP}^\infty _+) to degenerate at E2E_2. By the Kronecker pairing, the AHSS for E(CP+)E_*(\mathbb {CP}^\infty _+) also has to degenerate at E2E_2.

Now since H(MU)H_*(MU) is generated as a ring by the image of H(Σ2MU(1))=H(Σ2CP)H_*(\Sigma ^{-2} MU(1)) = H_*(\Sigma ^{-2}\mathbb {CP}^\infty ), we know the AHSS for E(MU)E_*(MU) also has to degenerate at E2E_2, and so does that for E(MU)E^*(MU). This gives us a preferred element in E0(MU)E^0(MU) which is equivalently a map MUEMU \to E as desired. Playing the same game with E(MUMU)E^*(MU \wedge MU) shows that this map is a homotopy ring map.

Proof

Corollary 3.8

A even homotopy ring spectrum is always complex orientable.

Proof
The Atiyah–Hirzebruch spectral sequence for CP\mathbb {CP}^\infty degenerates at E2E_2 for degree reasons.
Proof

Example 3.9

Complex KK-theory is complex orientable.