COMPLEX BORDISM

John Rognes

March 1st 1999

Lecture 9. Complex bordism

9.1. Stably almost complex structures.

Let M^{n} be a closed smooth n-manifold.

An almost complex structure on M is an embedding M^{n} → ℝ^{n+2k} with normal bundle ν^{2k} ↓ M and an almost complex structure J on ν^{2k}, i.e., a reduction of the structure group of ν^{2k} as an (O(2k), ℝ^{2k}) fiber bundle to a (U(k), ℂ^{k}) fiber bundle via r: U(k) → O(2k).

The almost complex structure can be stabilized to the composite embedding M^{n} → ℝ^{n+2k} ⊂ ℝ^{n+2k+2} with normal bundle ν^{2k+2} ≅ ν^{2k} ⊕ ϵ_{ℝ}^{2} ↓ M, and almost complex structure J ⊕ J_{0} where J_{0} is the standard almost complex structure on r(ϵ_{ℂ}^{1}) = ϵ_{ℝ}^{2}.

Two almost complex structures are stably equivalent if they can be (repeatedly) stabilized to isotopic embeddings, with isomorphic normal bundles and isomorphic almost complex structures. This defines an equivalence relation.

A stably almost complex n-manifold M^{n} is a closed smooth n-manifold M that comes equipped with a stable equivalence class of almost complex structures.

A similar definition applies for compact smooth manifolds W with boundary. A stably almost complex structure on M extends canonically to one on W = M × I (with I = [0, 1]), such that its restriction to M ≅ M × 0 ⊂ W is the original stably almost complex structure on M. The restriction of the stably almost complex structure on W to M ≅ M × 1 ⊂ W is called the opposite stably almost complex structure, and we let −M denote M with this opposite stably almost complex structure.

9.2. Complex (co)bordism.

Let X be a topological space.

A singular stably almost complex n-manifold over X is a stably almost complex closed smooth n-manifold M and a continuous map f: M → X. Two such singular n-manifolds (f: M → X) and (g: N → X) are said to be stably almost complex bordant, here written (f: M → X) ∼_{U} (g: N → X), if there exists a stably almost complex compact smooth (n + 1)-manifold W and a continuous map h: W → X, with ∂W = M ∐(−N) as stably almost complex manifolds, and h|∂W = f ∐ g.

For example, with M = N as stably almost complex manifolds and h: M×I → X a homotopy from f to g, then (h: M × I → X) is a bordism from (f: M → X) to (g: M → X).

Typeset by AMS-TEX

Let Ω_{n}^{U}(X) be the set of isomorphism classes of singular stably almost complex n-manifold over X, up to bordism:

Ω_{n}^{U}(X) = {M^{n} stably almost complex, f: M → X}/ ∼_{U}

This set is naturally an abelian group, with sum induced by disjoint union of stably almost complex manifolds, and inverse induced by taking (f: M → X) to (f: − M → X), i.e., M with the opposite stably almost complex structure and the same map f.

Theorem (R. Thom, S.P. Novikov, J.W. Milnor). The functor

X ↦ Ω_{n}^{U}(X)

is an unreduced homology theory, called complex bordism.

We write Ω_{n}^{U} = Ω_{n}^{U}(∗) for the coefficient groups of this homology theory. Cartesian product of stably almost complex manifolds defines a graded ring structure on Ω_{∗}^{U}. The resulting ring is called the complex bordism ring.

The proof that Ω_{n}^{U}(X, ∗) ≅ Ω_{n+1}^{U}(ΣX, ∗) depends heavily on transversality for stably almost complex manifolds. To explain, one must give a geometric description of the relative complex bordism groups Ω_{n}^{U}(X, A) in terms of singular stably almost complex n-manifolds over X, with boundary over A.

Then the inverse of the suspension isomorphism takes the bordism class of a singular (n+1)-manifold f: M^{n+1} → X ×I relative to X ×∂I to the inverse image N^{n} = f^{−1}(X × 1/2) ⊂ M^{n+1} as a singular n-manifold over X ≅ X × 1/2. For this to make sense, one must deform f to a representative of its bordism class with is transversal to the ‘equatorial’ copy of X in X × I. For smooth manifolds this can always be achieved, by the implicit function theorem. Then

σ: Ω_{n}^{U}(X) → Ω_{n+1}^{U}(X × I, X × ∂I) ≅ Ω̃_{n+1}^{U}(ΣX_{+})

takes the bordism class of (f|N: N → X) to the bordism class of (f: M → X × I), suitably relativized.

9.3. The Thom spectrum MU.

Let EU(k) ↓ BU(k) be the principal U(k)-bundle over BU(k) = Gr_{k}(ℂ^{∞}). In one model EU(k) is the space of unitary k-frames (v_{1}, …, v_{k}) in ℂ^{∞}, and E(γ_{ℂ}^{k}) = EU(k) ×_{U(k)} ℂ^{k} is the total space of the canonical ℂ^{k}-bundle γ_{ℂ}^{k} over BU(k) = Gr_{k}(ℂ^{∞}). This base space is a CW-complex with respect to the decomposition into Schubert cells.

Let D(γ_{ℂ}^{k}) ⊃ S(γ_{ℂ}^{k}) be the associated unit disc and sphere fiber bundles, with D(γ_{ℂ}^{k}) = EU(k) ×_{U(k)} D^{2k} and S(γ_{ℂ}^{k}) = EU(k) ×_{U(k)} S^{2k−1}. Let

MU(k) = T h(γ_{ℂ}^{k}) = D(γ_{ℂ}^{k})/S(γ_{ℂ}^{k})

be the Thom complex of γ_{ℂ}^{k}. This is a pointed CW complex.

Adding a fixed line to a k-dimensional subspace V ⊂ ℂ^{∞} gives a new subspace V ⊕ ℂ ⊂ ℂ^{∞} ⊕ ℂ ≅ ℂ^{∞}, which defines a map i_{k}: BU(k) → BU(k + 1), and i_{k}^{∗}(γ_{ℂ}^{k}) ≅ γ_{ℂ}^{k} ⊕ϵ_{ℂ}^{1} ↓ BU(k). There is thus a bundle map î_{k}: γ_{ℂ}^{k} ⊕ϵ_{ℂ}^{1} → γ_{ℂ}^{k+1} covering i_{k}.

The induced map of disc bundles is D(î_{k}): D(γ_{ℂ}^{k}) × D^{2} → D(γ_{ℂ}^{k+1}), which restricts to a map of sphere bundles S(î_{k}): S(γ_{ℂ}^{k}) × D^{2} ∪ D(γ_{ℂ}^{k}) × S^{1} → S(γ_{ℂ}^{k+1}). Hence there is an induced map of Thom complexes

T h(î_{k}): Σ^{2}T h(γ_{ℂ}^{k}) = Σ^{2}MU(k) ⟶ T h(γ_{ℂ}^{k+1}) = MU(k + 1) .

In the standard CW structure, this is an embedding onto a subcomplex.

We get a Σ-spectrum (an Adams type spectrum) MU with even spaces MU_{2k} = MU(k), odd spaces MU_{2k+1} = ΣMU(k), and structure maps m_{2k} = 1, m_{2k+1} = T h(î_{k}): Σ^{2}MU(k) → MU(k + 1).

Example: For k = 0, we have D(γ_{ℂ}^{0}) = BU(0) = ∗, S(γ_{ℂ}^{0}) = ∅, and MU(0) = T h(γ_{ℂ}^{0}) = ∗/∅ = S^{0}. The composite structure maps S^{2k} = Σ^{2k}MU(0) → MU(k) give a map of spectra η: S → MU. Later we will see that MU is a ring spectrum, and η is its unit map.

Example: For k = 1, the sphere bundle S(γ_{ℂ}^{1}) = S(ℂ^{∞}) = S^{∞} ≃ ∗ is contractible, so the quotient map D(γ_{ℂ}^{1}) → T h(γ_{ℂ}^{1}) = U(1) is a homotopy equivalence. Also the bundle projection map π: D(γ_{ℂ}^{1}) → BU(1) is a homotopy equivalence, so MU(1) ≃ BU(1) ≃ ℂP^{∞}.

The homotopy equivalence ℂP^{∞} → MU(1) = MU_{2} determines a canonical cohomology class ω ∈ M̃U^{2}(ℂP^{∞}). Its restriction to the 2-skeleton S^{2} = ℂP^{1} ⊂ ℂP^{∞} is a class in M̃U^{2}(S^{2}) ≅ MU^{0} ≅ ℤ, which turns out to be a generator.

Theorem (Thom). There is a natural isomorphism

MU_{n}(X) := colim_{k} π_{n+2k}(MU(k) ∧ X_{+}) ≅ Ω_{n}^{U}(X) .

In particular π_{n}MU := colim_{k} π_{n+2k}MU(k) ≅ Ω_{n}^{U}.

Thus complex bordism is the homology theory MU_{∗} represented by the Thom spectrum MU. The associated represented cohomology theory MU^{∗} is called complex cobordism.

We pause to describe the maps in Thom’s theorem. A continuous map g as in the diagram

S^{n+2k} ⟶^{g} MU(k) ∧ X_{+} ⊃ BU(k) × X

can be taken to be fiberwise smooth over X, away from ∗, and transverse to the zero section BU(k) × X in D(γ_{ℂ}^{k}) × X, so M^{n} = g^{−1}(BU(k) × X) ⊂ ℝ^{n+2k} ⊂ S^{n+2k} is a closed smooth n-manifold. The map g induces a bundle map ĝ: ν(M^{n} ⊂ S^{n+2k}) → γ_{ℂ}^{k} covering pr_{1} ∘ g|M: M → BU(k), which defines an almost complex structure on M. The map f := pr_{2} ∘ g|M: M → X defines the singular manifold over X. Hence one map in Thom’s theorem takes the stable homotopy class of g to the stably almost complex bordism class of (f: M → X).

Conversely, given a singular almost complex n-manifold (f: M → X) with embedding M^{n} → ℝ^{n+2k} and almost complex normal bundle ν^{2k} ↓ M there is a classifying map ν^{2k}: M^{n} → BU(k) covered by the disc bundle map ν̂: D(ν^{2k}) → D(γ_{ℂ}^{k}). There is also a map fπ: D(ν^{2k}) → X, and an embedding D(ν^{2k}) ⊂ ℝ^{n+2k} as a tubular neighborhood. The product map ν̂ × fπ: D(ν^{2k}) → D(γ_{ℂ}^{k}) × X takes S(ν^{2k}) to S(γ_{ℂ}^{k}) × X, so by the Pontryagin–Thom construction one obtains a map

g: S^{n+2k} ⟶ T h(γ_{ℂ}^{k}) ∧ X_{+} = MU(k) ∧ X_{+} .

This association provides the inverse map in Thom’s theorem.

9.4. The complex bordism ring.

For a homology theory E_{∗}, we call its value at the one-point space ∗ its coefficient groups. This is a graded abelian group, and is also denoted E_{∗} = E_{∗}(∗). For a reduced homology theory Ẽ_{∗}, the coefficient groups are the value at S^{0}, so E_{∗} = Ẽ_{∗}(S^{0}). Likewise, for a cohomology theory E^{∗} its coefficient groups are the value at ∗ and are denoted E^{∗} = E^{∗}(∗), while in the reduced case E^{∗} = Ẽ^{∗}(S^{0}).

When E is a spectrum, the coefficient groups of the associated homology theory are E_{∗} = {E_{n}}_{n} with E_{n} = Ẽ_{n}(S^{0}) = colim_{k} π_{n+k}(E_{k}). This is also the spectrum homotopy group π_{n}(E). For the associated cohomology theory, the coefficient groups are E^{∗} = {E^{n}}_{n} with E^{n} = colim_{k} π_{k}(E_{n+k}) = π_{−n}(E). Hence the homology and cohomology coefficient groups E_{∗} and E^{∗} are related by the formula E^{n} = E_{−n} for all integers n.

In the case E = MU, the coefficient groups MU_{∗} = π_{∗}(MU) are isomorphic to the complex bordism groups Ω_{∗}^{U} by Thom’s theorem. In fact these admit a ring structure, forming a graded ring. The product of two complex bordism classes [M] and [N] is given by the bordism class of the product manifold M × N, with a canonically defined stably almost complex structure. Thus MU_{∗} ≅ Ω_{∗}^{U} is a graded commutative ring, called the complex bordism ring.

(ETC.)