Landweber exact formal group laws and smooth cohomology theories

Ulrich Bunke  Thomas Schick  Ingo Schröder

Moritz Wiethaup

November 19, 2007

Abstract

The main aim of this paper is the construction of a smooth (sometimes called differential) extension M̂U of the cohomology theory complex cobordism MU, using cycles for M̂U(M) which are essentially proper maps W → M with a fixed U(n)- structure and U(n)-connection on the (stable) normal bundle of W → M.

Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties.

Moreover, we show, using the Landweber exact functor principle, that R̂(M) := M̂U(M)⊗_{MU^{∗}}R defines a multiplicative smooth extension of R(M) := MU(M)⊗_{MU^{∗}} R whenever R is a Landweber exact MU^{∗}-module. An example for this construction is a new way to define a multiplicative smooth K-theory.

Contents

1 Introduction 2

2 The Landweber construction and smooth extensions 3 2.1 The Landweber construction . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Smooth cohomology theories . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Smooth extensions for Landweber exact formal group laws . . . . . . . . . 5

3 Normal G-structures and bordism theories 7 3.1 Representatives of the stable normal bundle . . . . . . . . . . . . . . . . . 7 3.2 G-structures and connections on the stable normal bundle . . . . . . . . . 8 3.3 A cycle model for MG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Power series and genera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The smooth extension of MU 17

4.1 Characteristic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 The smooth extension of MU . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Smooth MU-orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 The push-forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1 Introduction

Smooth (also called differentiable) extensions of generalized cohomology theories recently became an intensively studied mathematical topic with many applications ranging from arithmetic geometry to string theory. Foundational contributions are [CS85], [Bry93] (in the case of ordinary cohomology) and [HS05]. The latter paper gives among many other results a general construction of smooth extensions in homotopy theoretic terms. For cohomology theories based on geometric or analytic cycles there are often alternative models. This applies in particular to ordinary cohomology whose smooth extension has various different realizations ([CS85], [Gaj97], [Bry93], [DL05], [HS05]). The recent paper [SS] shows that all these realizations are isomorphic. An example of a cycle model of a smooth extension of a generalized cohomology theory is the model of smooth K-theory introduced in [BS07], see also [Fre00],[FH00]. The present paper contributes geometric models of smooth extensions of bordism theories, where the case of complex bordism theory MU is of particular importance. We expect that our model M̂U of the smooth extension of MU is uniquely isomorphic to the one given by [HS05] though we have not checked the details.

An advantage of geometric or analytic models is that they allow the introduction of additional structures like products, smooth orientations and integration maps with good properties. These additional properties are fundamental for applications. In [HS05] methods for integrating and multiplying smooth cohomology classes were discussed, but further work will be required in order to turns these ideas into constructions with good functorial properties.

In the case of smooth ordinary cohomology the product and the integration have been considered in various places (see e.g. [CS85], [DL05], [Bry93]) (here smooth orientations are ordinary orientations). The case of smooth K-theory discussed in detail in [BS07] shows that in particular the theory of orientations and integration is considerably more complicated for generalized cohomology theories.

In the present paper we construct a multiplicative extension of MU. Furthermore, we introduce the notion of a smooth MU-orientation and develop the corresponding theory of integration. The same ideas could be applied with minor modifications to other bordism theories.

For an MU^{∗}-module R one can try to define a new cohomology theory R^{∗}(X) := MU^{∗}(X)⊗_{MU^{∗}}R for finite CW -complexes X. By a famous result [Lan76] this construction works and gives a multiplicative complex oriented cohomology theory provided R is a ring over MU^{∗} which is in addition Landweber exact. In Theorem 2.5 we observe that by the same idea one can define a multiplicative smooth extension R̂(X) := M̂U(X) ⊗_{MU^{∗}} R of R. It immediately follows that this smooth extension admits an integration for smoothly MU-oriented proper submersions.

In this way we considerably enlarge the class of examples of generalized cohomology theories which admit multiplicative extensions and integration maps. The construction can e.g. be applied to Landweber exact elliptic cohomology theories [LRS95], [Fra92] and complex K-theory ^{1}.

In Section 2 we review the main result of Landweber [Lan76] and the definition of a smooth extension of a generalized cohomology theory. We state the main result asserting the existence of a multiplicative smooth extension of MU with orientations and integration. Then we realize the idea sketched above and construct a multiplicative smooth extension for every Landweber exact formal group law.

In Section 3 we review the standard constructions of bordism theories using homotopy theory on the one hand, and cycles on the other. Furthermore, we review the notion of a genus.

In Section 4 we construct our model of the multiplicative smooth extension of complex bordism. Furthermore, we introduce the notion of a smooth MU-orientation and construct the integration map.

2 The Landweber construction and smooth extensions

2.1 The Landweber construction

2.1.1 Let X ↦ MU^{∗}(X) denote the multiplicative cohomology theory called complex bordism. We fix an isomorphism MU^{∗}(ℂℙ^{∞}) ≅ MU^{∗}[[x]]. The Künneth formula then gives MU^{∗}(ℂℙ^{∞} × ℂℙ^{∞}) ≅ MU^{∗}[[x, y]].

The tensor product of line bundles induces an H-space structure μ : ℂℙ^{∞} × ℂℙ^{∞} → ℂℙ^{∞}. Under the above identifications the map μ^{∗} : MU^{∗}[[z]] → MU^{∗}[[x, y]] is determined by the element f(x, y) := μ^{∗}(z) ∈ MU^{∗}[[x, y]].

By a result of Quillen [Qui69] the pair (MU^{∗}, f) is a universal formal group law. This means that given a commutative ring R and a formal group law g ∈ R[[x, y]] there exists a unique ring homomorphism θ : MU^{∗} → R such that θ_{∗}(f) = g. 2.1.2 Let R be a ring over MU^{∗}. Then one can ask if the functor X ↦ MU^{∗}(X)⊗_{MU^{∗}} R is a cohomology theory on the category of finite CW -complexes. The result of Landweber [Lan76] determines necessary and sufficient conditions. A ring which satisfies these conditions is called Landweber exact. 2.1.3 Actually, Landweber shows a stronger result which is crucial for the present paper. For any space or spectrum X the homology MU_{∗}(X) has a structure of a comodule over the coalgebra MU_{∗}MU in MU^{∗}-modules. By duality, if X is finite, then MU^{∗}(X) ≅ MU_{∗}(S(X)) also has a comodule structure, where S(X) denotes the Alexander-Spanier dual (see [Ada74]) of X.

1It is an interesting problem to understand the relation with the model [BS07]

Let M be a finitely presented MU^{∗}-module which has the structure of a comodule over MU_{∗}MU, and consider a Landweber exact formal group law (R, g) (so that in particular R is a ring over MU^{∗}).

Theorem 2.1 (Landweber [Lan76] ) We have Tor_{i}^{MU^{∗}}(M, R) = 0 for all i ≥ 1.

2.2 Smooth cohomology theories

2.2.1 Let N be a ℤ-graded vector space over ℝ. We consider a generalized cohomology theory h with a natural transformation of cohomology theories c : h(B) → H(B, N). The natural universal example is given by N := h^{∗}⊗ℝ, where c is the canonical transformation. Let Ω(B, N) := Ω(B) ⊗_{ℝ} N, where Ω(B) denotes the smooth real differential forms on B. By dR : Ω_{d=0}(B, N) → H(B, N) we denote the de Rham map which associates to a closed form the corresponding cohomology class. To a pair (h, c) we associate the notion of a smooth extension ĥ. Note that manifolds in the present paper may have boundaries.

Definition 2.2 A smooth extension of the pair (h, c) is a functor B → ĥ(B) from the category of compact smooth manifolds to ℤ-graded groups together with natural transformations

1. R : ĥ(B) → Ω_{d=0}(B, N) (curvature)

2. I : ĥ(B) → h(B) (forget smooth data)

3. a : Ω(B, N)/im(d) → ĥ(B) (action of forms) .

These transformations are required to satisfy the following axioms:

1. The following diagram commutes

\xymatrix{ĥ(B) \ar[d]^{R} \ar[r]^{I} & h(B) \ar[d]^{c} \\ Ω_{d=0}(B, N) \ar[r]^{dR} & H(B, N)} .

2.

R ∘ a = d .  (1)

3. a is of degree 1.

4. The sequence

h(B) →^{c} Ω(B, N)/im(d) →^{a} ĥ(B) →^{I} h(B) → 0 .  (2)

is exact.

2.2.2 If h is a multiplicative cohomology theory, then one can consider a ℤ-graded ring R over ℝ and a multiplicative transformation c : h(B) → H(B, R). In this case it makes sense to talk about a multiplicative smooth extension ĥ of (h, c).

Definition 2.3 A smooth extension ĥ of (h, c) is called multiplicative, if ĥ together with the transformations R, I, a is a smooth extension of (h, c), and in addition

1. ĥ is a functor to ℤ-graded rings,

2. R and I are multiplicative,

3. a(ω) ∪ x = a(ω ∧ R(x)) for x ∈ ĥ(B) and ω ∈ Ω(B, R)/im(d).

2.2.3 The first goal of the present paper is the construction of a multiplicative smooth extension of the pair (MU, c), where c : MU^{∗}(A) → MU^{∗}(A) ⊗_{ℤ} ℝ ≅ H^{∗}(A, MUℝ) is the canonical natural transformation (see 3.4.7). The following theorem is a special case of Theorem 4.21 which gives a construction of multiplicative smooth extensions of more general pairs (MU, h).

Theorem 2.4 The pair (MU, c) admits a multiplicative smooth extension.

The existence of a smooth extension also follows from [HS05], but how to derive the multiplicative structure from this reference is not so obvious, see [Wie]. 2.2.4 We also introduce the notion of a smooth MU-orientation (Definition 4.27) of a proper submersion p : W → B and define a push-forward p_{!} : M̂U(W) → M̂U(B) which refines the integration map p_{!} : MU(W) → MU(B) (Definition 4.34). In Subsection 4.4 we show that integration is compatible with the structure maps a, R, I of the smooth extension, functorial, compatible with pull-back and the product. We refer to this subsection and Theorem 2.7 for further details.

2.3 Smooth extensions for Landweber exact formal group laws

2.3.1 If (R, g) is a Landweber exact formal group law, then we let R^{∗}(X) := MU^{∗}(X)⊗_{MU^{∗}} R denote the associated cohomology theory on finite CW -complexes. We consider the pair (R, c_{R}), where c_{R} : R → R ⊗_{ℤ} ℝ =: Rℝ is the canonical map.

Theorem 2.5 If (R, g) is a Landweber exact formal group law, then (R, c_{R}) has a multiplicative smooth extension R̂.

Proof. We start with Theorem 2.4 which states that (MU, c) has a multiplicative smooth extension. Since Ω^{k}(∗) = 0 for k ≠ 0, Ω^{0}(∗) ≅ ℝ, and MU^{odd} = 0, the natural map M̂U^{ev}(∗) → MU^{ev}(∗) is an isomorphism. Hence M̂U^{ev}(∗) ≅ MU^{∗}, and M̂U(B) is a MU^{∗}-module. If B is compact, then MU^{∗}(B) is finitely presented. We set R̂(B) := M̂U^{∗}(B) ⊗_{MU^{∗}} R and define the structure maps R, I, a by tensoring the corresponding structure maps for M̂U. Here we identify R^{∗}(B) ≅ MU^{∗}(B) ⊗_{MU^{∗}} R and Ω(B, Rℝ) ≅ Ω(B, MUℝ) ⊗_{MU^{∗}} R. The only non-trivial point to show is that the sequence

R(B) →^{c_{R}} Ω(B, Rℝ)/im(d) →^{a} R̂(B) →^{I} R(B) → 0

is exact. Let us reformulate this as the exactness of

0 → Ω(B, Rℝ)/c_{R}(R(B)) → R̂(B) → R(B) → 0 .  (3)

We start from the exact sequence

0 → Ω(B, MUℝ)/c(MU^{∗}(B)) → M̂U(B) → MU^{∗}(B) → 0 .

Tensoring by R gives

Tor_{1}^{MU^{∗}}(MU^{∗}(B), R) → (Ω(B, MUℝ)/c(MU^{∗}(B))) ⊗_{MU^{∗}} R → M̂U(B) ⊗_{MU^{∗}} R → → MU^{∗}(B) ⊗_{MU^{∗}} R → 0 .

Since the tensor product commutes with quotients we have

(Ω(B, MUℝ)/c(MU^{∗}(B))) ⊗_{MU^{∗}} R ≅ Ω(B, Rℝ)/c_{R}(R(B)) .

We conclude the exactness of (3) from Landweber’s Theorem 2.1 which states that Tor_{1}^{MU^{∗}}(MU^{∗}(B), R) ≅ 0. □

2.3.2 Let p : V → A be a proper submersion which is smoothly MU-oriented (see 4.27) by o_{p}. Recall that R̂(V) = M̂U(V) ⊗_{MU_{∗}} R.

Definition 2.6 We define the push-forward map p_{!} : R̂(V) → R̂(A) by p_{!}(x ⊗ z) := p_{!}(x) ⊗ z.

We must show that the integration is well-defined. Let u ∈ MU(∗) ≅ M̂U^{ev}(∗). We must show that p_{!}(x ∪ u) ⊗ z = p_{!}x ⊗ uz. This indeed follows from the special case of the projection formula Lemma 4.39, p_{!}(x ∪ u) = p_{!}(x) ∪ u. The following theorem states that the natural and expected properties of a push-forward hold true.

Theorem 2.7 The following diagram commutes:

\xymatrix{Ω(V, Rℝ)/im(d) \ar[d]^{∫_{V/A} A(o_{p})∧...} \ar[r]^{a} & R̂(V) \ar[d]^{p_{!}} \ar[r]^{I} \ar[rr]^{R} & R^{∗}(V) \ar[d]^{p_{!}} & Ω(V, Rℝ) \ar[d]^{∫_{V/A} A(o_{p})∧...} \\ Ω(A, Rℝ)/im(d) \ar[r]_{a} & R̂(A) \ar[r]_{I} \ar[rr]_{R} & R^{∗}(A) & Ω(A, Rℝ)}

Furthermore, we have the projection formula

p_{!}(p^{∗}x ∪ y) = x ∪ p_{!}y , x ∈ R̂(A) , y ∈ R̂(V) .

The push-forward is compatible with pull-backs, i.e. for a cartesian diagram

\xymatrix{W \ar[d]^{q} \ar[r]^{F} & V \ar[d]^{p} \\ B \ar[r]^{f} & A}

we have

q_{!} ∘ F^{∗} = f^{∗} ∘ p_{!} : R̂(V) → R̂(B) ,

where q is smoothly MU-oriented by f^{∗}o_{p}.

If : C → V is a second proper submersion with smooth MU-orientation o_{r}, then the composition s := p ∘ r has the composed orientation o_{s} := o_{p} ∘ o_{r} (see 4.32), and we have

s_{!} = p_{!} ∘ r_{!} : R̂(C) → R̂(A) .

Proof. This follows immediately by tensoring with id_{R} the corresponding results of the push-forward for the extension of (MU, c). These are all proven in Section 4.4. □

3 Normal G-structures and bordism theories

3.1 Representatives of the stable normal bundle

3.1.1 In the present paper we construct geometric models of smooth extensions of bordism cohomology theories associated to the families G(n) of classical groups like U(n), SO(n), Spin(n), or Spin^{c}(n). We use the notation MG(B) and are in particular interested in the case where B is a smooth manifold. A cycle for MG^{n}(B) is a proper smooth map W → B of relative dimension −n with a normal G-structure. The relations are given by bordisms.

Cycles for the smooth extension will have in addition a geometric normal G-structure. In order to make a precise definition we introduce a rather concrete version of the notion of the stable normal bundle. 3.1.2 Let X be a space or manifold. Then \underline{ℝ^{k}}_{X} denotes the trivial real vector bundle X × ℝ^{k} → X. Let f : A → B be a smooth map between manifolds.

Definition 3.1 A representative of the stable normal bundle of f is a real vector bundle N → X together with an exact sequence

0 → T A ⟶^{(df,α)} f^{∗}T B ⊕ \underline{ℝ^{k}}_{A} → N → 0 ,

where we fix only the homotopy class of the projection to N.

There is a natural notion of an isomorphism of representative stable normal bundles. For an integer l ∈ ℕ it is evident how to define the l-fold stabilization N(l) := N ⊕ \underline{ℝ^{l}}_{A} as representative of the stable normal bundle with corresponding short exact sequence. 3.1.3 Let q : C → B be a smooth map which is transversal to f. Then we have a cartesian diagram

C ×_{B} A ⟶^{Q} A
↓^{F}  ↓^{f}
C ⟶^{q} B

of manifolds. If

0 → T A ⟶^{(df,α)} f^{∗}T B ⊕ \underline{ℝ^{k}}_{A} ⟶^{u} N → 0

represents the stable normal bundle of f, then we define the pull-back representative of the stable normal bundle of F by

0 → T (C ×_{B} A) ⟶^{(dF,β)} F^{∗}T C ⊕ \underline{ℝ^{k}}_{C×_{B}A} ⟶^{γ} Q^{∗}N → 0,

with β := Q^{∗}α ∘ dQ and γ := Q^{∗}u ∘ (F^{∗}dq ⊕ id\underline{ℝ^{k}}_{C×_{B} A}). Note that Q^{∗}(N(l)) ≅ (Q^{∗}N)(l). 3.1.4 We now discuss the stable normal bundle of a composition. Let g : B → C a smooth map and

0 → T B ⟶^{(dg,β)}^{←^{s}} g^{∗}T C ⊕ \underline{ℝ^{l}}_{B} ⟶^{v} M → 0  (4)

be a representative of the stable normal bundle of g. Then we define

0 → T A ⟶^{(d(gf),γ)} (gf)^{∗}T C ⊕ \underline{ℝ^{l}}_{A} ⊕ \underline{ℝ^{k}}_{A} ⟶^{w} N ⊕ f^{∗}M → 0

as the associated representative of the stable normal bundle of g∘f. Here γ := (f^{∗}β∘df, α) and w := (u ∘ f^{∗}s ⊕ id_{ℝ^{k}}, f^{∗}v ∘ pr_{(gf)^{∗}T C⊕0⊕\underline{ℝ^{k}}}), where s is the split indicated in (4). Note that this split is unique up to homotopy so that the homotopy class of w is well-defined.

3.2 G-structures and connections on the stable normal bundle

3.2.1 Let G be a Lie group with a homomorphism G → GL(n, ℝ) and consider an n-dimensional real vector bundle ξ → X.

Definition 3.2 A G-structure on ξ is a pair (P, ϕ) of a G-principal bundle P → X and an isomorphism of vector bundles ϕ : P ×_{G} ℝ^{n} →^{∼} ξ.

Definition 3.3 A geometric G-structure on ξ is a triple (P, ϕ, ∇), where (P, ϕ) is a Gstructure and ∇ is a connection on P.

Note that the trivial bundle \underline{ℝ^{n}}_{X} has a canonical G-structure with P = X × G → X.

3.2.2 In order to define a bordism theory we consider a sequence of groups G(n), n ≥ 0 which fit into a chain of commutative diagrams

\xymatrix{G(n) \ar[d] \ar[r] & GL(n, ℝ) \ar[d] \\ G(n + 1) \ar[r] & GL(n + 1, ℝ)} .

This is in particular used in order to define stabilization. In order to define the multiplicative structure we require in addition

\xymatrix{G(n) × G(m) \ar[d] \ar[r] & GL(n, ℝ) × GL(m, ℝ) \ar[d] \\ G(n + m) \ar[r] & GL(n + m, ℝ)} .

Examples are O(n), SO(n), Spin(n), or Spin^{c}(n). In the present paper we are in particular interested in the complex bordism theory MU. In this case we set G(2n) = G(2n+1) = U(n). By abuse of notation we will use the symbol G to denote such a family of groups, and by MG the corresponding bordism theory. 3.2.3 Let f : A → B be a smooth map between manifold.

Definition 3.4 A representative of a normal G-structure on f is given by a pair (N, P, ϕ), where N is a representative of the stable normal bundle, and (P, ϕ) is a G(n)-structure on N, where n := dim(N).

For notational convenience, we write N instead of the short exact sequence with quotient N which is also contained in the data of a representative of the stable normal bundle.

Definition 3.5 A representative of a geometric normal G-structure on f is given by a quadruple (N, P, ϕ, ∇), where N is a representative of the stable normal bundle of f, and (P, ϕ, ∇) is a geometric G(n)-structure on N, where n := dim(N).

There are natural notions of isomorphisms of representatives of normal G-structures or geometric normal G-structures. In the following we discuss the operations ”stabilization”, ”pull-back”, and ”composition” on the level of representatives of normal G-structure and geometric normal G-structures. 3.2.4 Let (N, P, ϕ) be a representative of a normal G-structure on f : A → B and consider l ∈ ℕ. The stabilization N(l) is N ⊕ \underline{ℝ^{l}}_{A}. It has a canonical G(n) × G(l)- structure with underlying principal bundle P × G(l) → A. We get a G(n + l)-structure with the underlying principal bundle

P (l) := (P × G(l)) ×_{G(n)×G(l)} G(l + n) .

Definition 3.6 We define the stabilization of (N, P, ϕ) by (N, P, ϕ)(l) := (N(l), P (l), ϕ(l)).

Let (N, P, ϕ, ∇) is a representative of a geometric normal G-structure, then the connection ∇ induces a connection ∇(l) on P (l).

Definition 3.7 We define the stabilization of (N, P, ϕ, ∇) by

(N, P, ϕ, ∇)(l) := (N(l), P (l), ϕ(l), ∇(l)) .

3.2.5 We now consider the pull-back and use the notation introduced in 3.1.3. If (P, ϕ) is a G(n)-structure on N, then (Q^{∗}P, Q^{∗}ϕ) is a G(n)-structure on Q^{∗}N.

Definition 3.8 We define the pull-back of a normal G-structure by

q^{∗}(N, P, ϕ) := (Q^{∗}N, Q^{∗}P, Q^{∗}ϕ) .

Definition 3.9 We define the pull-back of a geometric normal G-structure by

q^{∗}(N, P, ϕ, ∇) := (Q^{∗}N, Q^{∗}P, Q^{∗}ϕ, Q^{∗}∇) .

3.2.6 We now discuss the composition. Continuing with the notation of 3.1.4 we consider

A →^{f} B →^{g} C

and representatives of normal G-structures (N, P, ϕ) and (M, Q, ψ) on f and g. The sum N ⊕ f^{∗}M has a natural G(n) × G(m)-structure with underlying G(n) × G(m)-bundle P ×_{A} f^{∗}Q, and therefore a G(n + m)-structure with underlying bundle

R := (P ×_{A} f^{∗}Q) ×_{G(n)×G(m)} G(n + m)

with isomorphism ρ : R ×_{GL(n+m)} ℝ^{n+m} ≅ N ⊕ f^{∗}M.

Definition 3.10 We define the composition of representatives of normal G-structures by

(M, Q, ψ) ∘ (N, P, ϕ) := (N ⊕ f^{∗}M, R, ρ) .

If ∇^{P} and ∇^{Q} are connections on P and Q, then we get an induced connection ∇^{R} on R.

Definition 3.11 We define the composition of representatives of geometric normal Gstructures by

(M, Q, ψ, ∇) ∘ (N, P, ϕ, ∇) := (N ⊕ f^{∗}M, R, ρ, ∇^{R}) .

3.2.7 The following assertions are obvious.

Lemma 3.12 1. On the level of representatives of normal G-structures or geometric normal G-structures pull-back and composition commute with stabilization.

2. On the level of representatives of normal G-structures or geometric normal Gstructures pull-back and composition are functorial.

3. On the level of representatives of normal G-structures or geometric normal Gstructures pull-back and composition commute with each other.

3.3 A cycle model for MG

3.3.1 Let us fix a family of groups G as in 3.2.2. It determines a multiplicative cohomology theory which is represented by a Thom spectrum MG. The map G(n) → GL(n, ℝ) induces a map of classifying spaces BG(n) → BGL(n, ℝ). Let ξ_{n} → BG(n) denote the pull-back of the universal ℝ^{n}-bundle. Then we define MG^{n} := BG(n)^{ξ_{n}}, where for a vector bundle ξ → X we write X^{ξ} for its Thom space. The family of spaces MG^{n}, n ≥ 0, fits into a spectrum with structure maps

ΣMG^{n} ≅ BG(n)^{ξ_{n}⊕\underline{ℝ}_{BG(n)}} → BG(n + 1)^{ξ_{n+1}} ≅ MG^{n+1} ,

where we use the canonical cartesian diagram

ξ_{n} ⊕ \underline{ℝ}_{BG(n)} ⟶  ξ_{n+1}
↓  ↓
BG(n) ⟶ BG(n + 1) .

The ring structure is induced by

MG^{n}∧MG^{m} ≅ BG(n)^{ξ_{n}}∧BG(m)^{ξ_{m}} ≅ (BG(n)×BG(m))^{ξ_{n}⊞ξ_{m}} → BG(n+m)^{ξ_{n+m}} ≅ MG^{n+m} ,

using the canonical cartesian diagram

ξ_{n} ⊞ ξ_{m}  ⟶  ξ_{n+m}
↓  ↓
BG(n) × BG(m) ⟶ BG(n + m) .

If A is a manifold (or more generally a finite CW-complex), then the homotopy theoretic definition of the bordism cohomology group is

MG^{n}(A) := lim_{k}[Σ^{k}A_{+}, MG^{n+k}] ,

where limit is taken over the stabilization maps

[Σ^{k}A_{+}, MG^{n+k}] → [ΣΣ^{k}A_{+}, ΣMG^{n+k}] → [Σ^{k+1}A_{+}, MG^{n+k+1}] ,

and A_{+} is the union of A and a disjoint base point. Temporarily we use the bold-face notation of the homotopy theoretic definition of the bordism cohomology theory. For details we refer to [Swi02] or [Sto68]. 3.3.2 We now present a cycle model of the G-bordism theory. Let A be a smooth manifold.

Definition 3.13 A precycle (p, ν) of degree n ∈ ℤ over A consists of a smooth map p : W → A with n = dim(A) − dim(W), and a representative ν of a normal G-structure on p (see 3.4). A cycle of degree n ∈ ℤ over A is a precycle (p, ν) of degree n, where p is proper.

There is a natural notion of an isomorphism of precycles.

3.3.3 Let c := (p, ν) be a precycle over A and q : B → A be transverse to p.

Definition 3.14 We define the pull-back q^{∗}c := (q^{∗}p, q^{∗}ν), a precycle over B (using 3.8).

The pull-back is functorial by Lemma 3.12. 3.3.4 We now consider precycles c = (p, ν) over A and d = (q, μ) over C with underlying maps p : B → A and q : A → C.

Definition 3.15 We define the composition

d ∘ c := (q ∘ p, μ ∘ ν)

using 3.10.

The composition d∘c is a precycle over C. The composition is associative and compatible with pull-back. 3.3.5 Let c := (p, ν), p : W → A, and d := (q, μ), q : V → B be precycles over A and B. Then we can form the diagram

\xymatrix{ & W × V \ar[d]^{Q} \ar[r] & V \ar[d]^{q} \\ W \ar[d]^{p} & W × B \ar[l] \ar[d]^{P} \ar[r]^{r} & B \\ A & A × B \ar[l]^{s}}.

Definition 3.16 We define the product of the precycles c and d to be the precycle c×d := s^{∗}c ∘ r^{∗}d over A × B.

Note that there is an equivalent definition based on on the diagram

\xymatrix{ & W × V \ar[d] \ar[r] & W \ar[d]^{p} \\ V \ar[d]^{q} & A × V \ar[l] \ar[d] \ar[r] & A \\ B & A × B \ar[l]} .

It follows from the functoriality of the composition and its compatibility with the pull-back that the product of precycles is associative. 3.3.6 We consider a precycle b := ((f, p), ν) over ℝ × A.

Definition 3.17 The precycle b is called a bordism datum if f is transverse to {0} ∈ ℝ and p_{|{f≥0}} is proper. We define the precycle ∂b := i^{∗}b, where i : A → ℝ×A, i(a) := (0, a).

3.3.7 Let c = (p, ν) be a precycle and l ∈ ℕ.

Definition 3.18 We define the l-fold stabilization of the precycle c by c(l) := (p, ν(l)) (see 3.6).

3.3.8 We now come to the geometric picture of the bordism theory MG. We consider a smooth manifold A and let ZMG(A) denote the semigroup of isomorphism classes of cycles over A with respect to disjoint union. Recall that a relation ∼ on a semigroup is compatible with the semigroup structure if a ∼ b implies that a + c ∼ b + c for all c.

Definition 3.19 Let “∼” be the minimal equivalence relation which is compatible with the semigroup structure and satisfies:

1. If b is a bordism datum, then ∂b ∼ 0.

2. If l ∈ ℕ, then c(l) ∼ c.

We let MG(A) := ZMG(A)/ ∼ denote the quotient semigroup.

3.3.9 Let 0 denote the cycle of degree n based on the empty manifold. The following Lemma will be useful in calculations.

Lemma 3.20 Let c be a cycle which is equivalent to 0. Then there exists a bordism datum b and l ≥ 0 such that c(l) ≅ ∂b.

We leave the proof to the interested reader. 3.3.10 We now describe the functoriality, the product, orientations, and the integration on the level of cycles.

1. Functoriality Let f : B → A be a smooth map and x ∈ MG(A). We can represent x by a cycle c = (p, ν) such that p and f are transverse. Then f^{∗}x is represented by f^{∗}c.

2. Product Let c and d be cycles for x ∈ MG^{n}(A) and y ∈ MG(B)^{m}. Then x × y ∈ MG^{n+m}(A × B) is represented by the cycle c × d (see 3.16). We get the interior product using the pull-back along the diagonal.

3. Integration Let d be a cycle over A with underlying map q : V → A. In this situation we have an integration map

q_{!} : MU(V) → MU(A) .

If x ∈ MG(V) is represented by the cycle c, then q_{!}(x) is represented by the cycle d ∘ c (see 3.15).

4. Suspension Let i : ∗ → S^{1} denote the embedding of a point. The trivial bundle \underline{ℝ}_{∗} represents the stable normal bundle which of course has a canonical G(1)-structure. In this way i is the underlying map of a cycle {i} ∈ ZMG^{1}(S^{1}) which represents a class [i] ∈ MG^{1}(S^{1}). For a manifold A we define MG^{k}(A) → MG^{k+1}(S^{1} × A), x ↦ {i} × x, which on the level of cycles is represented by c ↦ {i} × c. This transformation is essentially the suspension morphism (not an isomorphism, since we do not reduce).

3.3.11 In order to show that the cycle level operations defined above descend through the equivalence relation ∼ the following observations are useful. Let b = ((f, p), μ) be a bordism datum over A with underlying map (f, p) : W → ℝ × A. Assume that q : B → A is transverse to p and p_{|{f=0}}. Then we can form the bordism datum (id_{ℝ} × q)^{∗}b over B which will be denoted by q^{∗}b. Note that

q^{∗}∂b ≅ ∂q^{∗}b .

Let e be a cycle over B. Then we can form b × e which we can interpret as a bordism datum over B × A. Note that

∂(b × e) ≅ ∂b × e .

Let d be a cycle with underlying map A → B. Let pr : ℝ × B → B be the projection. Then we can form the bordism datum pr^{∗}d ∘ b over B. Note that

∂(pr^{∗}d ∘ b) ≅ d ∘ ∂b .

Finally, if c is a cycle over W, then we can form the bordism datum b ∘ c over B, and have

∂(b ∘ c) ≅ ∂b ∘ c .

3.3.12 We now have a geometric and a homotopy theoretic picture of the G-bordism theory which we distinguish at the moment by using roman and bold-face letters.

Proposition 3.21 There is an isomorphism of ring-valued functors MG(A) ≅ MG(A) on compact manifolds. This isomorphism preserves the product and is compatible with push-forward.

Proof. This follows from the well-known Pontryagin-Thom construction. We have to observe that upto stabilisation and homotopy the normal bundle sequence

0 → T M → f^{∗}T A ⊕ \underline{R^{n}}_{M} → N → 0

comes from an embedding of M into A × ℝ^{n}. □

In view of Proposition 3.21 we can drop the bold-face notation for the homotopy theoretic bordism.

3.4 Power series and genera

3.4.1 The basic datum for a multiplicative smooth extension of a generalized cohomology theory h is a pair (h, c), where c : h → HR is a natural transformation from h into the ordinary cohomology with coefficients in a graded ring R over ℝ. The transformation c induces in particular a homomorphism of coefficients h^{∗} → R^{∗}. Our construction of smooth extensions of bordism theories is based on a description of c in terms of characteristic numbers of stable normal bundles.

A ring homomorphisms c : MG → R is called a G-genus. One can classify SO and Ugenera in terms of formal power series (see [HBJ92] and 3.22). Genera for other bordism theories can be derived from transformations like MSpin → MSO. Since the details in the real and complex case differ slighty, in the present paper we restrict to our main example MU. It is easy to modify the constructions for other cases like MSpin^{c}, MSO or Spin^{c}. 3.4.2 Let R be a commutative ℤ-graded algebra over ℝ with 1 ∈ R^{0}. By R[[z]] we denote the graded ring of formal power series, where z has degree 2. Let ϕ ∈ R[[z]]^{0} be a power series of the form 1 + ϕ_{1}z + ϕ_{2}z^{2} + … (note that deg(ϕ_{i}) = −2i). To such a power series we associate a genus r_{ϕ} : MU^{∗} → R^{∗}.

Theorem 3.22 ([HBJ92]) The correspondence ϕ → r_{ϕ} gives a bijection between the set R[[z]]^{0} and R-valued U-genera.

In the following we describe the associated natural transformation r_{ϕ} : MU(A) → H(A, R) of cohomology theories on the level of cycles. 3.4.3 We define the power series K_{ϕ} ∈ R[[σ_{1}, σ_{2}, …,]]^{0} (where σ_{i} has degree 2i) such that

K_{ϕ}(σ_{1}, σ_{2}, …) = ∏_{i=1}^{∞} ϕ(z_{i})

holds if we replace σ_{i} by the elementary symmetric functions σ_{i}(z_{1}, …). 3.4.4 Note that G(2k) = G(2k +1) = U(k) (see 3.2.2). Let N → W be an n-dimensional real vector bundle with a G(n)-structure (P, ϕ). Then we have Chern classes c_{j}(N) := c_{j}(P) ∈ H^{2j}(W, ℝ).

Definition 3.23 We define the characteristic class

ϕ(N) := K_{ϕ}(c_{1}(N), c_{2}(N), …) ∈ H^{0}(A, R) .

The following properties are well-known (see[HBJ92]).

Lemma 3.24 1. Let \underline{ℝ^{k}}_{A} have the trivial G(k)-structure. Then we have ϕ(\underline{ℝ^{k}})_{A} = 1 for all k ≥ 0.

2. If M is a second bundle with a G(m)-structure, and N⊕M has the induced G(n+m)- structure, then we have ϕ(N ⊕ M) = ϕ(N) ∪ ϕ(M).

3. If f : B → A is a continuous map, then we have f^{∗}ϕ(N) = ϕ(f^{∗}N), if we equip f^{∗}N with the induced G(n)-structure.

3.4.5 Consider a cycle c = (p, ν) ∈ ZMU(A) of degree n with underlying map p : W → A and normal U-structure ν = (N, P, ϕ). Then p is a proper map which is oriented for the ordianary cohomology theory HR. In particular we have an integration p_{!} : H^{∗}(W, R) → H^{∗−n}(A, R).

Definition 3.25 We define

r̃_{ϕ}(c) := p_{!}(ϕ(N)) ∈ H^{n}(A, R) .

3.4.6 The following Lemma implies half of Theorem 3.22. What is missing is the argument that every R-valued U-genus comes from a formal power series.

Lemma 3.26 The map r̃_{ϕ} descends through ∼ and induces a natural transformation r_{ϕ} : MU(A) → H(A, R) of ring-valued functors.

Proof. Using the first and second property in 3.24 one checks that r̃_{ϕ}(c) = r̃_{ϕ}(c(l)).

Assume that b = ((f, q), μ) with underlying map q : V → A and μ = (M, Q, λ) is a bordism datum. Then we get the diagram

\xymatrix{V \ar[r]^{i} \ar[d]^{p} & W \ar[d]^{(f,q)} \\ \{0\} × A \ar[r] & ℝ × A}

We have N ≅ i^{∗}M. Therefore p_{!}(ϕ(N)) = p_{!}(ϕ(i^{∗}M)) = p_{!}(i^{∗}ϕ(M)) = 0 by the bordism invariance of the push-forward in ordinary cohomology and the third property of 3.24. Thus the transformation r_{ϕ} is well-defined.

It is natural since for f : B → A which is transverse to p we have a cartesian diagram

\xymatrix{F^{∗}N \ar[d] \ar[r] & N \ar[d] \\ f^{∗}V \ar[d]^{q} \ar[r]^{F} & V \ar[d]^{p} \\ B \ar[r]^{f} & A} ,

the bundle F^{∗}N represents the stable normal bundle of q, and

q_{!}(ϕ(f^{∗}N)) = q_{!}(F^{∗}ϕ(N)) = f^{∗}p_{!}(ϕ(N))

by the projection formula. This implies that f^{∗}r̃_{ϕ}(c) = r̃_{ϕ}(f^{∗}c).

The transformation is also multiplicative.

To this end we consider a cycle d = (q, μ) with underlying map q : V → B and normal G-structure μ = (M, Q, λ). Then the underlying proper map of c × d ∈ ZMU(A × B) is p × q : W × V → A × B, and the bundle N ⊞ M represents its normal G-structure. We thus have

(p × q)_{!}(ϕ(N ⊞ M)) = (p × q)_{!}(ϕ(N) × ϕ(M)) = p_{!}(ϕ(N)) × q_{!}(ϕ(M)) .

This implies

r̃_{ϕ}(c × d) = r̃_{ϕ}(c) × r̃_{ϕ}(d) .

3.4.7 The most important example for the present paper is given by the ring MUℝ := MU^{∗} ⊗_{ℤ} ℝ. The MU^{∗}-module MUℝ is Landweber exact. Hence, for a compact manifold or finite CW -complex A we have H^{∗}(A, MUℝ) ≅ MU^{∗}(A) ⊗_{MU^{∗}} MUℝ and therefore a canonical natural transformation r : MU^{∗}(A) → H^{∗}(A, MUℝ), x ↦ x ⊗ 1. This transformation is a genus r = r_{ϕ} for a certain power series ϕ ∈ MUℝ[[x]]^{0}. We refer to [HBJ92] for further details on ϕ.

4 The smooth extension of MU

4.1 Characteristic forms

4.1.1 Let ϕ ∈ R[[z]]^{0} be as in 3.4.2 and G be the family of groups 3.2.2 associciated to U(n), n ≥ 0.. We first lift the construction of the characteristic class ϕ(N) ∈ H^{0}(A, R) of vector bundles N → A with G(n)-structure to the form level.

Let (P, ψ, ∇^{N}) be a geometric G(n)-structure on N → A. By R^{∇^{N}} ∈ Ω^{2}(A, End(N)) we denote the curvature of the connection ∇^{N}. The fibre-wise polynomial bundle morphism det : End(N) → \underline{ℝ}_{A} extends to det : Ω^{ev}(A, End(N)) → Ω^{ev}(A). As usual we define the Chern forms c_{i}(∇^{N}) ∈ Ω^{2i}(A) by

1 + c_{1}(∇^{N}) + c_{2}(∇^{N}) + ⋯ = det(1 + \frac{1}{2πi}R^{∇^{N}}) .

Definition 4.1 If N → A is a real vector bundle with a geometric G(n)-structure, then we define

ϕ(∇^{N}) := K_{ϕ}(c_{1}(∇^{N}), c_{2}(∇^{N}), …) ∈ Ω^{0}(A, R) .

4.1.2 The properties stated in Lemma 3.24 lift to the form level.

Lemma 4.2 1. Let k ≥ 0 and \underline{ℝ^{k}}_{A} have the trivial G(k)-structure with the trivial connection. Then we have ϕ(∇^{\underline{ℝ^{k}}^{A}}) = 1.

2. If M → A is a second bundle with a geometric G(m)-structure and assume that N ⊕ M has the induced geometric G(n + m)-structure, then we have ϕ(∇^{N⊕M}) = ϕ(∇^{N}) ∧ ϕ(∇^{M}).

3. If f : B → A is a smooth map. Then we have f^{∗}ϕ(∇^{N}) = ϕ(∇^{f^{∗}N}), if we equip f^{∗}N with the induced geometric G(n)-structure.

4.1.3

Definition 4.3 A geometric precycle over A is a pair (p, ν) of a smooth map p : V → A and a geometric normal G-structure ν (see 3.5). A geometric precycle is a cycle if p is proper.

Usually we will denote geometric precycles by c̃, where c denotes the underlying precycle. Since a principal bundle always admits connections every precycle can be refined to a geometric precycle. If ν = (N, P, ϕ, ∇), then we will write ∇^{ν} := ∇.

4.1.4 Let Ω_{−∞}(A) := C^{−∞}(A, Λ^{∗}T^{∗}A) denote the differential forms with distributional coefficients. We identify this space with the topological dual of C_{c}^{∞}(A, Λ^{n−∗}T^{∗}A ⊗ Λ_{A}), where Λ_{A} → A is the real orientation bundle and n = dim(A). For this identification we use cup product and integration of an n-form with values in the orientation bundle over A. Finally, we define Ω_{−∞}(A, R) := Ω_{−∞}(A) ⊗_{ℝ} R using the algebraic tensor product.

A morphism of complexes inducing an isomorphism in cohomology is called a quasiisomorphism. It is well-known (see [dR84], or do this exercise using Lemma 4.11) that the inclusion Ω(A) ↪ Ω_{−∞}(A) is a quasi-isomorphism. Hence, Ω(A, R) ↪ Ω_{−∞}(A, R) is a quasi-isomorphism, too. 4.1.5 Let p : V → A be a proper smooth oriented map. The orientation of p gives an isomorphism p^{∗}Λ_{A} →^{∼} Λ_{V}. We then define the push-forward

p_{!} : Ω_{−∞}(V) → Ω_{−∞}(A)

of degree dim(A) − dim(V) such that

< p_{!}ω, σ >=< ω, p^{∗}σ > , ω ∈ Ω_{−∞}(V) , σ ∈ Ω(A, Λ_{A})

holds true. By tensoring with the identity of R we get the map p_{!} : Ω_{−∞}(V, R) → Ω_{−∞}(A, R). Stoke’s theorem implies

p_{!} ∘ d = d ∘ p_{!} .

The following diagram commutes

\begin{matrix}H^{∗}(Ω_{−∞}(V, R)) & ⟶_{≅}^{deRham} & H^{∗}(V, R) \\ ↓^{p_{!}} & & ↓^{p_{!}} \\ H^{∗}(Ω_{−∞}(A, R)) & ⟶_{≅}^{deRham} & H^{∗}(A, R) . \end{matrix}  (5)

4.1.6 Let c̃ = (p, ν) be a geometric cycle of degree n.

Definition 4.4 We define T (c̃) := p_{!}(ϕ(∇^{ν})) ∈ Ω_{−∞}^{n}(A, R).

This form is closed, and by (5) we have the following equality in de Rham cohomology:

[T (c̃)] = p_{!}(ϕ(N)) = r̃_{ϕ}(c) .  (6)

4.1.7 We now consider a bordism datum b = ((f, q), μ) over a manifold A with (f, q) : W → ℝ × A. We build the composition

q_{!} ∘ χ_{{f≥0}} : Ω^{k}(W) → Ω_{−∞}^{k+l}(A) ,

where l := dim(A) − dim(W), and χ_{U} is the multiplication operation with the characteristic function of the subset U. Stoke’s theorem implies in this case that

d ∘ q_{!} ∘ χ_{{f≥0}} − q_{!} ∘ χ_{{f≥0}} ∘ d = (q_{0})_{!} ∘ i^{∗} ,  (7)

where q_{0} : W_{0} → A is defined by the cartesian diagram

\begin{matrix}W_{0} & →^{i} & W \\ q_{0} ↓ & & q ↓ \\ A & →^{a↦(0,a)} & ℝ × A \end{matrix} ,

i.e. q_{0} is the underlying map of ∂b.

Definition 4.5 Let b̃:= ((f, q), ν̃) be a geometric refinement of b. We define

T (b̃) := q_{!} ∘ χ_{{f≥0}}(ϕ(∇ν̃)) ∈ Ω_{−∞}(A)

Equation (7) shows that

dT (b̃) = T (∂b̃) .  (8)

4.2 The smooth extension of MU

4.2.1 In the present subsection we construct the smooth extension associated to the pair (MU, r_{ϕ}), where ϕ ∈ R[[z]]^{0} is as in 3.4.2, and r_{ϕ} is the associated natural transformation MU(A) → H(A, R). Recall the notions of a cycle and a geometric cycle from 3.13 and 4.3. The cycles for the smooth extension M̂U of MU will be called smooth cycles.

Definition 4.6 A smooth cycle of degree n is a pair ĉ := (c̃, α), where c̃ is a geometric cycle of degree n, and α ∈ Ω_{−∞}^{n−1}(A, R)/im(d) is such that

T (c̃) − dα := Ω(ĉ) ∈ Ω^{n}(A, R) .

The point here is that T (c̃) − dα is a smooth representative of the cohomology classes represented by T (c̃). The latter is in general a singular form. To be explicit note that in the definition above

im(d) := im(d : Ω_{−∞}^{n−2}(A, R) → Ω_{−∞}^{n−1}(A, R)) ,

i.e. we allow differentials of forms with distribution coefficients. 4.2.2 There is a natural notion of an isomorphism of smooth cycles. We form the graded semigroup ZM̂U(A) of isomorphism classes of smooth cycles such that the sum is given by

(c̃, α) + (c̃′, α′) = (c̃ + c̃′, α + α′) ,

where, as in the non-geometric case, c̃ + c̃′ is given by the disjoint union.

4.2.3 The smooth bordism group M̂U(A) will be defined as the quotient of ZM̂U(A) by an equivalence relation generated by stabilization and bordism.

Definition 4.7 Let “∼” be the minimal equivalence relation on ZM̂U(A) which is compatible with the semigroup structure (see 3.3.8) and such that

1. For l ∈ ℕ we have (c̃, α) ∼ (c̃(l), α), where c̃(l) is the l-fold stabilization defined by (p, ν)(l) := (p, ν(l)) (see 3.7).

2. For a geometric bordism datum b̃ we have (∂b̃, T (b̃)) ∼ 0.

We define M̂U^{n}(A) := ZM̂U^{n}(A)/ ∼ as the semigroup of equivalence classes of smooth cycles of degree n.

We will write [c̃, α] for the equivalence class of (c̃, α). 4.2.4

Lemma 4.8 M̂U^{n}(A) is a group.

Proof. Let [c̃, α] ∈ M̂U(A). It suffices to show that it admits an inverse. Since MU(A) is a group there exists a cycle c′ such that c + c′ ∼ 0. By Lemma 3.20 we can assume that c(l) + c′(l) ∼ ∂b for some bordism datum b and l ∈ ℕ. We extend b to a geometric bordism datum b̃ by choosing a connection such that ∂b̃ ≅ c̃(l) + c̃′(l) for some geometric extension c̃′ of c′. Then we have [c̃′, T (b̃) − α] + [c̃, α] = 0. □

4.2.5 We now define the structure maps a, R, I (see 2.2) of the smooth extension M̂U.

Definition 4.9 1. We define R : M̂U(A) → Ω_{d=0}(A, R) by R([c̃, α]) := T (c̃) − dα.

2. We define a : Ω(A, R) → M̂U(A) by a(α) := [∅, −α].

3. We define I : M̂U(A) → MU(A) by I([c̃, α]) := [c] (using the geometric model 3.19)

Lemma 4.10 These maps are well-defined. We have R ∘ a = d.

Proof. The only non-obvious part is the fact that R is well-defined. To this end consider a geometric bordism datum b̃. Then we have

R[∂b̃, T (b̃)] = T (∂b̃) − dT (b̃) = 0

by Equation (8). □

4.2.6 We now extend A ↦ M̂U(A) to a contravariant functor on the category of smooth manifolds. Let f : B → A be a smooth map. Then we must construct a functorial pull-back f^{∗} : M̂U(A) → M̂U(B) such that the transformations R, I, a above become natural.

Let (c̃, α) be a smooth cycle with c̃ = (p, ν), p : W → A. We can assume that p is transverse to f. Otherwise we replace p by a bordant (homotopic) map and correct α correspondingly. Then we have the cartesian diagram

\xymatrix{B ×_{A} W \ar[d]^{P} \ar[r]^{F} & W \ar[d]^{p} \\ B \ar[r]^{f} & A} .

The map P is the underlying map of a geometric cycle f^{∗}c̃ = (P, f^{∗}ν), where f^{∗}ν is the pull-back of the geometric normal G-structure as defined in 3.9. We want to define f^{∗}[c̃, α] := [f^{∗}c̃, f^{∗}α]. The problem is that α is a distribution. In order to define the pullback f^{∗}α of a distributional form we need the additional assumption that W F (α)∩N(f) = ∅, where N(f) ⊆ T^{∗}A \ 0_{A} is the normal set to f given by

N(f) := clo{η ∈ T^{∗}A \ 0_{A} | ∃b ∈ B s.t. f(b) = π(η) and df(b)^{∗}η = 0}

(where π : T^{∗}A → A is the projection), and W F (α) denotes the wave front set of α (see [Hör03]). Note that we consider α up to exact forms with distribution coefficients. The idea is to show that one can choose α such that W F (α) ∩ N(f) = ∅ holds. In this case f^{∗}α is defined and is independent of the choice again up to exact forms with distribution coefficients. The details will be explained in the following paragraphs. 4.2.7

Lemma 4.11 Let α ∈ Ω_{−∞}^{n}(A). Then there exists β ∈ Ω_{−∞}^{n−1}(A) such that W F (α−dβ) ⊆ W F (dα).

Proof. We choose a Riemannian metric on A. Then we can define the formal adjoint δ := d^{∗} of the de Rham differential and the Laplacian Δ := δd + dδ. Since Δ is elliptic we can choose a proper pseudo-differential parametrix P of Δ. Then we form G := δP. This pseudo-differential operator satisfies dG + Gd = 1 + S, where S is a proper smoothing operator. We thus can set β := Gα and have

α − dβ = Gdα − Sα .

Since Sα is smooth and W F (Gdα) ⊆ W F (dα) (a pseudo-differential operator does not increase wave front sets) we see that W F (α − dβ) ⊆ W F (dα). □

If α ∈ Ω_{−∞}(A, R), then we can write for some s ∈ ℕ

α = ∑_{i=1}^{s} α_{i} ⊗ r_{i}

with α_{i} ∈ Ω_{−∞}^{n}, and with linearly independent r_{i} ∈ R. In this case the wave front set of α is by definition W F (α) := ∪_{i=1}^{s}W F (α_{i}). It is now easy to see that Lemma 4.11 extends to forms with coefficients in R. 4.2.8

Lemma 4.12 If (c̃, α), c̃ = (p, ν), is a smooth cycle, then we can choose α such that W F (α) ⊆ N(p).

Proof. It is a general fact that the wave front set of the push-forward of a smooth distribution along a map is contained in the normal set of the map. In view of the definition 4.4 we have W F (T (c̃)) ⊆ N(p). Since T (c̃) − dα is smooth we have W F (dα) = W F (T (c̃)) ⊆ N(p), and by Lemma 4.11 we can change α by an exact form such that W F (α) ⊆ N(p). □

4.2.9 A reformulation of the fact that f and p are transverse is that N(f) ∩ N(p) = ∅. Using Lemma 4.12 we now take a representative of α such that W F (α) ⊆ N(p). Then f^{∗}α is a well-defined distribution.

Definition 4.13 We define f^{∗}[c̃, α] = [f^{∗}c̃, f^{∗}α], where we take representatives c̃ = (p, ν) and α such that f and p are transverse and W F (α) ⊆ N(p).

4.2.10

Lemma 4.14 The pull-back is well-defined and functorial.

Proof. First we show that the pull-back is well-defined with respect to the choice of α. Let β ∈ Ω_{−∞}(A, R) and α′ := α + β be such that T (c̃) − dα′ is smooth. This implies that W F (α′) ⊆ N(p), and hence W F (dβ) ⊆ N(p). By Lemma 4.11 we can modify β by a closed form such that W F (β) ⊆ N(p). Then f^{∗}α′ = f^{∗}α + df^{∗}β.

It is easy to see that the pull-back is additive and preserves stabilization. It remains to show that it preserves zero bordism. Let b̃ = ((h, q), μ) be a geometric bordism datum over A with (h, q) : W → ℝ × A. We define W_{0} := h^{−1}({0}) and assume that q and q_{|W_{0}} are transverse to f. We then have the geometric bordism datum (id_{ℝ} × f)^{∗}b̃ over B.

Let us define the normal datum of b by

N(b) := clo{η ∈ T^{∗}A \ 0_{A} | ∃v ∈ W s.t. E(v) = π(η) and [dE(v)^{∗}η = 0
or v ∈ W_{0} and dE(v)^{∗}η_{|T_{v}W_{0}} = 0]}.

Then we have W F (T (b̃)) ⊆ N(b). Again, since q and q_{|W_{0}} are transverse to f we have N(b)∩N(f) = ∅ so that f^{∗}T (b̃) is well defined. Using the fact that in a cartesian diagram push-forward of distributions commutes with pull-back we get f^{∗}T (b̃) = T (f^{∗}b̃). It follows that (f^{∗}∂b̃, f^{∗}T (b̃)) = (∂f^{∗}b̃, T (f^{∗}b̃)). This implies that the pull-back is well-defined on the level of equivalence classes.

We now show functoriality. Let g : C → B be a second smooth map. If x̂ ∈ M̂U(A), then we can choose the representing smooth cycle (c̃, α) with c̃ = (p, ν) such that p is transverse to f and f ∘ g. In this case one easily sees that (f ∘ g)^{∗}(c̃, α) and g^{∗}f^{∗}(c̃, α) are isomorphic cycles. □

4.2.11 We now have defined a functor A ↦ M̂U(A) from smooth manifolds to graded groups.

Lemma 4.15 The transformations R, I and a are natural.

Proof. Straightforward. □

4.2.12 We now define the outer product

× : M̂U(A) ⊗ M̂U(B) → M̂U(A × B) .

Let x̂ ∈ M̂U(A) be represented by (c̃, α), and let ŷ ∈ M̂U(B) be represented by (ẽ, β). In 3.16 we have already defined the product of cycles c × e. Here we enhance this definition to the geometric level. Write c̃ = (p, ν) and d̃ = (q, μ). Then we define c̃ × d̃:= (p × q, ν ⊕ μ), where the sum of geometric normal G structures ν ⊕ μ is defined similarly as in the non-geometric case.

Note that we have a graded outer product

× : Ω_{−∞}(A, R) ⊗ Ω_{−∞}(B, R) → Ω_{−∞}(A × B, R) .

Definition 4.16 We define the product of smooth cycles (c̃, α) × (ẽ, β) by

(c̃ × ẽ, (−1)^{|x̂|}R(x̂) × β + α × T (e)) ,

and we define the product x̂ × ŷ ∈ M̂U(A × B) to be the corresponding equivalence class.

This cycle level definition needs a few verifications.

Lemma 4.17 1. The outer product is well-defined.

2. It is associative, i.e. (x̂ × ŷ) × ẑ = x̂ × (ŷ × ẑ), where ẑ ∈ M̂U(C).

3. It is graded commutative in the sense that F^{∗}(x̂ × ŷ) = (−1)^{|x̂||ŷ|}ŷ × x̂, where F : B × A → A × B is the flip F (b, a) := (a, b).

4. The product is natural, i.e. if f : C → A is a smooth map, then we have f^{∗}x̂ × ŷ = (f × id_{B})^{∗}(x̂ × ŷ).

Proof. We first show that the cycle level definition of the outer product passes through the equivalence relation. It is obvious that the outer product is bilinear and preserves stabilizations in both arguments. It remains to verify that it preserves zero bordisms.

Let b̃ be a geometric bordism datum. Then we can form the geometric bordism datum b̃ × ẽ (see 3.16). We have T (b̃ × ẽ) = T (b̃) × T (ẽ) so that

(∂b̃, T (b̃)) × (ẽ, β) = (∂b̃ × ẽ, T (b̃) × T (ẽ))
= (∂(b̃ × ẽ), T (b̃ × ẽ))
∼ 0 .

In order to see that the product also preserves zero bordism in the second entry we rewrite

(−1)^{|x̂|}R(x̂) × β + α × T (ẽ) ≡^{im(d)} (−1)^{|x̂|}T (c̃) × β + α × R(ŷ)  (9)

and apply the same argument as above. Associativity, graded commutativity, and naturality hold true on the level of smooth cycles. For commutativity we use again (9), and the proof of associativity is based on similar calculations. □

4.2.13 As usual, the outer product determines a graded commutative ring structure by restriction to the diagonal.

Definition 4.18 We define the ring structure on M̂U(A) by x̂ ∪ ŷ := Δ^{∗}(x̂ × ŷ), where Δ : A → A × A is the diagonal.

The following assertions are consequences of Lemmas 4.14 and 4.17.

Corollary 4.19 A ↦ M̂U(A) is a contravariant functor from the category of manifolds to the category of graded rings.

Lemma 4.20 The transformations R and I are multiplicative, and we have a(α) ∪ x̂ = a(α ∧ R(x̂)).

Proof. Straightforward calculation. □

4.2.14 Recall that we have fixed a graded ring R over ℝ and a formal power series ϕ ∈ R[[z]]^{0} which determines an R-valued U-genus r_{ϕ} 3.22, 3.26, 3.4.7.

Theorem 4.21 The functor M̂U together with the transformations R, I, a is a multiplicative smooth extension of the pair (MU, r_{ϕ}).

Proof. We must verify the properties required in Definitions 2.2 and 2.3. Most of them have been shown above. We are left with the commutativity of

\xymatrix{M̂U(B) \ar[d]^{R} \ar[r]^{I} & MU(B) \ar[d]^{r_{ϕ}} \\ Ω_{d=0}(B, R) \ar[r]^{dR} & H(B, R)} .  (10)

and the exactness of

MU(B) →^{r_{ϕ}} Ω(B, R)/im(d) →^{a} M̂U(B) →^{I} MU(B) → 0 .  (11)

The commutativity of the diagram (10) is a direct consequence of (6).

We now discuss exactness of (11). We start with the surjectivity of I. Let x ∈ MU(B) be represented by a cycle c. Then we can choose a geometric refinement c̃. We have dT (c̃) = 0, and by Lemma 4.11 there exists α ∈ Ω_{−∞}(B, R) such that T (c̃) − dα is smooth. Therefore (c̃, α) is a smooth cycle, and we have x = I[c̃, α].

We now discuss exactness at M̂U(B). It is clear that I ∘ a = 0. Let x̂ ∈ M̂U(B), be such that I(x̂) = 0. Then we can assume that x̂ is of the form [∂b̃, α] for some geometric bordism datum b̃. Hence x̂ = a(T (b̃) − α).

We now show exactness at Ω(B, R)/im(d). Let x ∈ MU(B) be represented by a cycle c. Then we choose a geometric refinement c̃, and by 4.11 a form α ∈ Ω_{−∞}(B, R) such that T (c̃) − dα is smooth. We have r_{ϕ}(x) = T (c̃) − dα. Let c = (p, ν) with p : V → B, and consider the constant map h : V → ℝ with value 1. The geometric normal U-structure of (h, p) : V → ℝ × B can also be represented by ν. Then b̃ = ((h, p), ν) is a geometric bordism datum with ∂b̃ = ∅ and T (b̃) = T (c̃). It follows

a(dα − T (c̃)) = [∂b̃, T (b̃) − dα] = [∂b̃, T (b̃)] = 0 .

This proves that a ∘ r_{ϕ} = 0.

Let now α ∈ Ω(A, R) be such that a(α) = 0. Then there exist geometric bordism data b̃_{0}, b̃_{1} such that ∂b̃_{0} ≅ ∂b̃_{1} and T (b̃_{0}) − T (b̃_{1}) − α ∈ im(d). This already implies that α is closed, We construct a geometric cycle c̃ such that T (c̃) = T (b̃_{0}) − T (b̃_{1}) by glueing the bordism data along their common boundary. Then [α] = [T (c̃)] = r_{ϕ}([c]) in de Rham cohomology. □

4.3 Smooth MU-orientations

4.3.1 As before we fix a graded ring R over ℝ and a formal power series ϕ ∈ R[[z]]^{0}. Let M̂U be the smooth extension of (MU, r_{ϕ}) as in Theorem 4.21 with structure maps R, a, I. If q : V → A is a proper MU-oriented map, then we have an integration q_{!} : MU(V) → MU(A) (see 3.3.10). Under the assumption that q is a submersion we introduce the notion of a smooth MU-orientation and define the integration map q_{!} : M̂U(V) → M̂U(A). The reason of the restriction to submersions is that otherwise one can not expect a construction which satisfies a projection formula, see [Wie] 4.3.2 Let q : V → A be a proper submersion.

Definition 4.22 A representative of a smooth MU-orientation of q is a pair č:= (c̃, σ), where c̃ is a geometric cycle with underlying map q : V → A and σ ∈ Ω^{−1}(V, R).

A representative of a smooth MU-orientation of q induces in particular an MU-orientation of q.

4.3.3 We now introduce an equivalence relation ∼ called stable homotopy on the set of representatives of smooth MU-orientations of q.

Definition 4.23 We define the l-fold stabilization of (c̃, σ) by (c̃, σ)(l) := (c̃(l), σ).

Let h_{i} : A → ℝ×A denote the inclusions h_{i}(a) := (i, a), i = 0, 1. Consider a geometric cycle d̃= (p, μ) over ℝ × A with underlying map p := id_{ℝ} × q : ℝ × V → ℝ × A. It gives rise to a closed form ϕ(∇^{μ}) ∈ Ω^{0}(ℝ × V, R). Let c̃_{i} := h_{i}^{∗}d̃, c̃_{i} = (q, ν_{i}).

Definition 4.24 We call d̃ a homotopy between c̃_{0} and c̃_{1}.

Definition 4.25 We define the transgression form

ϕ̃(∇^{ν_{1}}, ∇^{ν_{0}}) := ∫_{[0,1]×V/V} ϕ(∇^{μ}) ∈ Ω^{−1}(A, R)/im(d) .

Since the underlying cycle d of d̃is a product, and since the space of geometric refinements of d is contractible, the transgression form is well-defined indepently on the choice of the homotopy (this is a standard argument in the theory of characteristic forms). By Stoke’s theorem the transgression satisfies

dϕ̃(∇^{ν_{1}}, ∇^{ν_{0}}) = ϕ(∇^{ν_{1}}) − ϕ(∇^{ν_{0}}) .  (12)

Definition 4.26 We call two representatives of a smooth MU-orientation (c̃_{i}, σ_{i}) homotopic if there exists a homotopy d̃ from c̃_{0} to c̃_{1}, and σ_{1} − σ_{0} = ϕ̃(∇^{ν_{1}}, ∇^{ν_{0}}).

4.3.4 We now define equivalence of representatives of smooth MU-orientations.

Definition 4.27 Let ∼ be the minimal equivalence relation on the set of representatives of smooth MU-orientations on q such that

1. (c̃, σ) ∼ (c̃(l), σ)

2. (c̃_{0}, σ_{0}) ∼ (c̃_{1}, σ_{1}), if (c̃_{0}, σ_{0}) and (c̃_{1}, σ_{1}) are homotopic.

A smooth MU-orientation of q is an equivalence class of representatives of smooth MUorientations which we will usually write as o := [c̃, σ].

4.3.5 Let c̃:= (q, ν) and č:= (c̃, σ) be a representative of a smooth MU-orientation.

Definition 4.28 We define A(č) := ϕ(∇^{ν}) − dσ ∈ Ω^{0}(V, R).

Lemma 4.29 The form A(č) only depends on the smooth MU-orientation [č] represented by č.

Proof. This immediately follows from (12) and the definition of homotopy. □

Below we will write A(o) := A(č), where o := [č].

4.3.6 In the following two paragraphs we define the operations of pull-back and composition of smooth MU-orientations. We start with the pull-back. Let f : B → A be a smooth map which is transverse to q. Then we have the cartesian diagram

\xymatrix{W \ar[d]^{Q} \ar[r]^{F} & V \ar[d]^{q} \\ B \ar[r]^{f} & A} .

Definition 4.30 We define the pull-back of a representative of a smooth MU-orientation of q by f^{∗}(c̃, σ) := (f^{∗}c̃, F^{∗}σ) (see 4.2.6) which is a representative of a smooth MUorientation of Q.

Lemma 4.31 The pull-back is compatible with the equivalence relation. It induces a functorial pull-back of smooth MU-orientations. We have A(f^{∗}o) = F^{∗}A(o).

Proof. It is clear that the pull-back is compatible with stabilization. Let d̃be a homotopy from c̃_{0} to c̃_{1}. Then (id_{ℝ} × f)^{∗}d̃ is a homotopy from f^{∗}c̃_{0} to f^{∗}c̃_{1}. Furthermore, one checks that ϕ̃(∇^{f^{∗}ν_{1}}, ∇^{f^{∗}ν_{0}}) = f^{∗}ϕ̃(∇^{ν_{1}}, ∇^{ν_{0}}). These formulas imply that the pull-back preserves homotopic representatives of smooth MU-orientations. We conclude that the pull-back is well-defined on the level equivalence classes. Functoriality and the relation of characteristic forms is easy to see. □

4.3.7 We now define the composition. Let p : A → B be a second proper submersion, and let (d̃, θ), d̃ = (p, μ), be a representative of a smooth MU-orientation of p. Let o_{q} = [c̃, σ] and o_{p} := [d̃, θ]. By d̃∘ c̃ we denote the composition of geometric cycles which is based on Definition 3.11.

Definition 4.32 We define

o_{p} ∘ o_{q} := [d̃∘ c̃, A(o_{q}) ∧ q^{∗}θ + σ ∧ q^{∗}ϕ(∇^{μ})] .

The definition requires some verifications.

Lemma 4.33 The composition of smooth MU-orientations is well-defined, compatible with pull-back, and functorial.

Proof. We first show that the composition is well-defined. It is clear that the composition is compatible with stabilization. Let b̃ be a homotopy from c̃_{0} to c̃_{1}. Then pr_{2}^{∗}d̃∘ b̃ is a homotopy from b̃ ∘ c̃_{0} to b̃ ∘ c̃_{1}, where pr_{2} : ℝ × B → B is the projection. We further calculate (using the properties stated in Lemma 4.2)

(σ_{1} − σ_{0}) ∧ q^{∗}ϕ(∇^{μ}) = ϕ̃(∇^{ν_{1}}, ∇^{ν_{0}}) ∧ q^{∗}ϕ(∇^{μ})
= ϕ̃(∇^{μ∘ν_{1}}, ∇^{μ∘ν_{0}}) .

This calculation implies that the composition (d̃, θ) ∘ … preserves homotopic representatives.

Let us now consider a homotopy ẽ from d̃_{0} to d̃_{1} We get a homotopy ẽ ∘ c̃ from d̃_{0} ∘ c̃ to d̃_{1} ∘ c̃. Furthermore we rewrite (note that we work modulo im(d))

A(o_{q}) ∧ q^{∗}θ + σ ∧ q^{∗}ϕ(∇^{μ}) = ϕ(∇^{ν}) ∧ q^{∗}θ + σ ∧ q^{∗}A(o_{p}) .

We have

ϕ(∇^{ν}) ∧ q^{∗}(θ_{1} − θ_{0}) = ϕ(∇^{ν}) ∧ q^{∗}ϕ̃(∇^{μ_{1}}, ∇^{μ_{0}})
= ϕ̃(∇^{μ_{1}∘ν}, ∇^{μ_{0}∘ν}) .

Hence ⋯ ∘ (c̃, ν) preserves homotopic representatives. This finishes the proof that the composition is well-defined. 4.3.8 The composition of smooth MU-orientations is associative and compatible with pull-back. For completeness let us state the second fact in greater detail. Let r : Q → B be a map which is transverse to q and p ∘ q. Then we have the composition of pull-back diagrams

\xymatrix{Q ×_{B} V \ar[r] \ar[d] & V \ar[d]^{q} \\ Q ×_{B} A \ar[d] \ar[r]^{s} & A \ar[d]^{p} \\ Q \ar[r]^{r} & B} .

Then we have

s^{∗}o_{p} ∘ r^{∗}o_{q} = r^{∗}(o_{p} ∘ o_{q}) .

The details of the straightforward proof we leave to the reader. □

4.4 The push-forward

4.4.1 Let p : V → A be a proper submersion with a smooth MU-orientation o_{p} := [d̃, σ], d̃= (p, ν). In the following (c̃, α) denotes a smooth cycle on V, and we use the notation

∫_{V/A} := p_{!} : Ω_{−∞}(V, R) → Ω_{−∞}(A, R)

for the integration of forms.

Definition 4.34 We define the push-forward on the level of cycles by

p_{!}(c̃, α) = (d̃∘ c̃, ∫_{V/A}(ϕ(∇^{ν}) ∧ α + σ ∧ R(c̃, α))) .

Lemma 4.35 For fixed (d̃, σ) the push-forward preserves equivalence of smooth cycles. Furthermore, the induced map p_{!} : M̂U(V) → M̂U(A) only depends on the equivalence class [d̃, σ] of representatives of the smooth MU-orientation.

Proof. It is clear that the push-forward is additive and compatible with stabilization. Let now b̃ be a geometric bordism datum over V. Let pr : ℝ × A → A be the projection and form (ẽ, θ) := pr^{∗}(d̃, σ). Then ẽ ∘ b̃ is a bordism datum, and we have T (ẽ ∘ b̃) = ∫_{V/A} ϕ(∇^{ν}) ∧ T (b̃). We calculate

p_{!}(∂b̃, T (b̃)) = (d̃∘ ∂b̃, ∫_{V/A} ϕ(∇^{ν}) ∧ T (b̃))
= (∂(ẽ ∘ b̃), T (ẽ ∘ b̃)) .

This equality implies that p_{!} perserves zero bordisms.

For a fixed representative (d̃, σ) of the smooth MU-orientation we now have a welldefined map p_{!} : M̂U(V) → M̂U(A). Next we show that it only depends on the smooth orientation represented by (d̃, σ). Again it is clear that stabilization of the representative of the smooth orientation does not change p_{!}. We now consider a homotopy b̃ from (d̃_{0}, σ_{0}) to (d̃_{1}, σ_{1}). The idea of the argument is to translate this homotopy into a bordism datum. To this end we first consider a model case. Let κ : ℝ → ℝ be defined by κ(x) := x − x^{2}. Then κ^{−1}({[0, ∞)}) = [0, 1]. We choose a representative of the stable normal bundle of κ with a geometric U-structure μ such that κ̃ = (κ, μ) is a geometric bordism datum.

Let pr_{1} : ℝ × A → ℝ denote the projection. The composition r̃ := pr_{1}^{∗}κ̃ ∘ b̃ is now a bordism datum. Let ρ denote the representative of the geometric U-structure on the normal bundle of r. We consider r̃ ∘ pr_{2}^{∗}c̃ as a geometric bordism datum with ∂(r̃∘pr_{2}^{∗}c̃) = d̃_{0} ∘c̃+(d̃_{1} ∘c̃)^{op}, where ()^{op} indicates a flip of the orientation. Let c̃ = (q, ν), q : U → V and d̃_{i} = (p, λ_{i}).

T (r̃ ∘ pr_{2}^{∗}c̃) = ∫_{q^{−1}r^{−1}([0,∞)×V)/A} ϕ(∇^{ρ}) ∧ ϕ(∇^{ν})
= ∫_{V/A} (ϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ ∫_{U/V} ϕ(∇^{ν}))

On the other hand

∫_{V/A}(ϕ(∇^{λ_{1}}) − ϕ(∇^{λ_{0}})) ∧ α + (σ_{1} − σ_{0}) ∧ R(c̃, α)
= ∫_{V/A} dϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ α + ϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ R(c̃, α)
= ∫_{V/A} ϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ dα + ϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ R(c̃, α)
= ∫_{V/A} (ϕ̃(∇^{λ_{1}}, ∇^{λ_{0}}) ∧ ∫_{U/V} ϕ(∇^{ν}))

These two equations together show that (d̃_{1}, σ_{1}) ∘ (c̃, α) ∼ (d_{0}, σ_{0}) ∘ (c̃, α). Indeed (d̃_{0}∘c̃+(d̃_{1}∘c̃)^{op}, ∫_{V/A}(ϕ(∇^{λ_{1}})−ϕ(∇^{λ_{0}}))∧α+(σ_{1}−σ_{0})∧R(c̃, α)) = (∂(r̃∘pr_{2}^{∗}c̃), T (r̃∘pr_{2}^{∗}c̃)) .

□

4.4.2 Lemma 4.36 The following diagram commutes.

\xymatrix{Ω(V, R)/im(d) \ar[d]^{∫_{V/A} A(o_{p})∧...} \ar[r]^{a} & M̂U(V) \ar[d]^{p_{!}} \ar[r]^{I} \ar[rr]^{R} & MU(V) \ar[d]^{p_{!}} & Ω(V, R) \ar[d]^{∫_{V/A} A(o_{p})∧...} \\ Ω(A, R)/im(d) \ar[r]_{a} & M̂U(A) \ar[r]_{I} \ar[rr]_{R} & MU(A) & Ω(A, R)}

Proof. Commutativity of the left square follows from partial integration

∫_{V/A}(ϕ(∇^{ν}) ∧ α − σ ∧ dα) = ∫_{V/A}(ϕ(∇^{ν}) − dσ) ∧ α = ∫_{V/A} A(o_{p}) ∧ α .

For the right square we use

T (d̃∘ c̃) = ∫_{V/A} ϕ(∇^{ν}) ∧ T (c̃)

which implies

R(p_{!}(c̃, α)) = T (d̃∘ c̃) − d ∫_{V/A}(ϕ(∇^{ν}) ∧ α + σ ∧ R(c̃, α))
= ∫_{V/A}(ϕ(∇^{ν}) ∧ T (c̃) − ϕ(∇^{ν}) ∧ dα − dσ ∧ R(c̃, α))
= ∫_{V/A}(ϕ(∇^{ν}) − dσ) ∧ R(c̃, α)
= ∫_{V/A} A(o_{p}) ∧ R(c̃, α) .

Commutativity of the middle square is a direct consequence of geometric description of p_{!} : MU(V) → MU(A) (see 3.3.10). □

4.4.3 Let p : V → A be as before with the smooth MU-orientation o_{p} := [d̃, σ]. We furthermore consider a proper submersion q : A → B with a smooth MU-orientation o_{q} := [ẽ, ρ], ẽ = (q, μ). Let r := q ∘ p : V → B be equipped with the composed smooth MU-orientation o_{r} := o_{q} ∘ o_{p} (see Definition 4.32)

Lemma 4.37 The push-forward ist functorial, i.e. we have the equality

r_{!} = q_{!} ∘ p_{!} : M̂U(V) → M̂U(B) .

Proof. The equality holds on the smooth cycle level. The proof is a straightforward calculation of both sides by inserting the definitions und using the right square in Lemma 4.36. □

4.4.4 Let p : V → A be a proper smoothly MU-oriented map as above, and let f : B → A be a second smooth map so that we get a cartesian diagram

\xymatrix{W \ar[d]^{P} \ar[r]^{F} & V \ar[d]^{p} \\ B \ar[r]^{f} & A} .

The map P has an induced smooth MU-orientation o_{P} := f^{∗}o_{p} (see Definition 4.30).

Lemma 4.38 The push-forward commutes with pull-back, i.e. we have the equality

P_{!} ∘ F^{∗} = f^{∗} ∘ p_{!} : M̂U(V) → M̂U(B) .

Proof. The equality holds true on the level of smooth cycles (c̃, α) whose underlying map is transverse to F. By definition we have o_{P} = [f^{∗}d̃, f^{∗}σ]. Furthermore, it follows immediately from the definitions that f^{∗}(d̃∘ c̃) = f^{∗}d̃∘ F^{∗}c̃. The final ingredient of the verification is the identity

f^{∗} ∘ ∫_{V/A} … = ∫_{W/B} ∘F^{∗} ….
□

4.4.5 Let p : V → A be as above a smoothly MU-oriented proper submersion.

Lemma 4.39 The projection formula holds true, i.e. for x ∈ M̂U(A) and y ∈ M̂U(V) we have p_{!}(p^{∗}x ∪ y) = x ∪ p_{!}y.

Proof. We consider the diagram

\xymatrix{V \ar[d]^{p} \ar[r]^{(p,id_{)}} \ar[rr]^{^{id^{V}}} & A × V \ar[r]^{^{pr^{2}}} \ar[d]^{q} & V \ar[d]^{p} \\ A \ar[r]^{Δ} \ar[rr]_{_{id_{A}}} & A × A \ar[r]^{_{pr_{2}}} & A} ,

where q := id_{A} × p has the induced orientation o_{q} := pr_{2}^{∗}o_{p}. If we show that

q_{!}(x × y) = x × p_{!}(y) ,  (13)

then by the definition of the cup-product and applying Lemma 4.38 to the left cartesian square we get the result. Equation (13) holds true on the level of smooth cycles and is straightforward to check. □

References

[Ada74] J. F. Adams. Stable homotopy and generalised homology. University of Chicago Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics.

[Bry93] Jean-Luc Brylinski. Loop spaces, characteristic classes and geometric quantization, volume 107 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993.

[BS07] Ulrich Bunke and Thomas Schick. Smooth K-theory, 2007, arXiv:0707.0046

[CS85] Jeff Cheeger and James Simons. Differential characters and geometric invariants. In Geometry and topology (College Park, Md., 1983/84), volume 1167 of Lecture Notes in Math., pages 50–80. Springer, Berlin, 1985.

[DL05] Johan L. Dupont and Rune Ljungmann. Integration of simplicial forms and Deligne cohomology. Math. Scand., 97(1):11–39, 2005.

[dR84] Georges de Rham. Differentiable manifolds, volume 266 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984. Forms, currents, harmonic forms, Translated from the French by F. R. Smith, With an introduction by S. S. Chern.

[FH00] Daniel S. Freed and Michael Hopkins. On Ramond-Ramond fields and K-theory. J. High Energy Phys., (5):Paper 44, 14, 2000, arXiv: hep-th/0002027.

[Fra92] Jens Franke. On the construction of elliptic cohomology. Math. Nachr., 158:43– 65, 1992.

[Fre00] Daniel S. Freed. Dirac charge quantization and generalized differential cohomology. In Surveys in differential geometry, Surv. Differ. Geom., VII, pages 129–194. Int. Press, Somerville, MA, 2000.

[Gaj97] Paweł Gajer. Geometry of Deligne cohomology. Invent. Math., 127(1):155–207, 1997.

[HBJ92] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung. Manifolds and modular forms. Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum.

[Hör03] Lars Hörmander. The analysis of linear partial differential operators. I. Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)].

[HS05] M. J. Hopkins and I. M. Singer. Quadratic functions in geometry, topology, and M-theory. J. Differential Geom., 70(3):329–452, 2005.

[Lan76] Peter S. Landweber. Homological properties of comodules over MU_{∗}(MU) and BP_{∗}(BP). Amer. J. Math., 98(3):591–610, 1976.

[LRS95] Peter S. Landweber, Douglas C. Ravenel, and Robert E. Stong. Periodic cohomology theories defined by elliptic curves. In The Čech centennial (Boston, MA, 1993), volume 181 of Contemp. Math., pages 317–337. Amer. Math. Soc., Providence, RI, 1995.

[Qui69] Daniel Quillen. On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc., 75:1293–1298, 1969.

[SS] James Simons and Dennis Sullivan. Axiomatic Characterization of Ordinary Differential Cohomology, arXiv:math.AT/0701077.

[Sto68] Robert E. Stong. Notes on cobordism theory. Mathematical notes. Princeton University Press, Princeton, N.J., 1968.

[Swi02] Robert M. Switzer. Algebraic topology—homotopy and homology. Classics in Mathematics. Springer-Verlag, Berlin, 2002. Reprint of the 1975 original [Springer, New York].

[Wie] Moritz Wiethaup. PhD-thesis, Göttingen, 2007.