A geometric description of smooth cohomology

In this paper we give a geometric cobordism description of smooth integral cohomology. This model allows for simple descriptions of both the cup product and the integration, so that it is easy to verify the compatibilty of these structures.


Axioms of smooth cohomology theories
To begin, let us recall what is meant by a smooth extension of the functor H k , ordinary cohomology. Compare [1, Definition 1.1] for the formal definition and the fundamental paper [6] for a general construction of such theories. We denote the closed k-forms by Ω k cl (X) and the map to de Rham cohomology, which we identify with real singular cohomology via the de Rham isomorphism, by Rham : Ω k cl (X) → H k (X; R).

Definition 1.
A smooth extension of H is a functor X →Ĥ * (X) from the category of smooth manifolds to Z-graded groups together with natural transformations 3. a : Ω * −1 (X)/im(d) →Ĥ * (X) (action of forms).
These transformations have to satisfy the following axioms.

For every smooth manifold X the sequence
is exact, where the first map is the composition To have a compatible ring structure means thatĤ actually takes values in graded commutative rings (we denote the product by ∪), that R, I are ring maps, and that for all x ∈Ĥ(X) and ω ∈ Ω(X)/im(d) we have a(ω) ∪ x = a(ω ∧ R(x)).
In this case we call the smooth extension a multiplicative smooth extension.
We will use a construction of H k (X) in terms of cobordism classes similar to Quillen's description of singular cobordism. The difference is that we replace manifolds by manifolds with singularities called stratifolds. In the next section we briefly introduce stratifolds and prove some basic properties needed for our construction. To distinguish the cohomology theory constructed from stratifolds from singular cohomology we denote it by SH k (X).

Stratifolds
Now we give a short introduction to stratifolds, where we are rather sketchy and refer the reader to [9] for details. A stratifold is a topological space S together with a subsheaf C of the sheaf of continuous functions, which in case of a smooth manifold plays the role of the sheaf of smooth functions. The space S and the sheaf C have to fulfill certain natural axioms, which in particular give a decomposition of S into smooth manifolds, the strata of S. The top-dimensional stratum S m is also called the regular part S reg . There is an obvious definition of morphisms between stratifolds, which are continuous maps f : S → S ′ , which pull elements in C ′ back to elements in C. A basic property which relates the strata S k to the stratifold S is that for each x ∈ S k there is an open neighborhood U ⊆ S of x and a retract r : U → U ∩ S k which is a morphism. These are called local retracts. It is useful to note that a map is a morphism if and only if it is smooth on all strata and commutes with appropriate local retracts. We will only consider regular stratifolds which means that locally near each x in a stratum S k the stratifold looks like V × F for some stratifold F and open subset V in S k . A stratifold is called orientable if the codimension 1 stratum is empty and the top stratum is orientable. Once an orientation on the top stratum is fixed we call such a stratifold an oriented stratifold. More generally, if S is a stratifold and X a smooth manifold, a continuous map f : is the orientation covering of X. An orientation of f is then the choice of such an isomorphism. Note that it also gives an isomorphism between Λ S reg and f | * S ref Λ X , where Λ X is the real orientation bundle of X.
We also consider stratifolds with boundary. This is a pair of spaces (T , ∂T ) together with stratifold structures on T −∂T and on ∂T and a germ of collars c : ∂T × [0, 1) → T . Many basic properties of smooth manifolds generalize to stratifolds, like tangent spaces (the vector space of derivations of the germ of morphisms to R), the differential of a morphism, differential forms (see below), Sard's theorem, approximation of continuous maps from a stratifold to a smooth manifold by morphisms and the transversality theorem for a map from a stratifold S to a smooth manifold X and a smooth map from a manifold Y → X. For all this see [9].
Since we will use differential forms intensively we define them on stratifolds. A k-form ω on a stratifold S is a prescription which to each x ∈ S assigns an alternating k-form on T x S, which fulfills the following property: 1. The restriction to each stratum is a differential k-form.

2.
For each x ∈ S r , the r-stratum, there is an open neighborhood U ⊆ S of x in S and a local retract Here the pull back of a differential form under a morphism from a stratifold to a smooth manifold is defined as for smooth manifolds using the differential.
such that the restriction of f to U factors over r and hence Note that U and r are determined by f and can be chosen independent of ω.
Since f * (ω)| U ∩S r = 0 for dimensional reasons we conclude that there is an open neighborhood V of the singular part S − S m such that (f | S m ) * (ω) vanishes on S m ∩ V . Now, let K be a compact set in X such that ω vanishes outside of K.
is compact, and f * (ω) vanishes outside this set.
As a consequence we can define the following integral. Let ω be a form with compact support on X with coefficients in Λ X and f : S → X a proper oriented morphism from an m-dimensional stratifold S to X. Then we define using the identification of f | * S m Λ X with Λ S m from the orientation of f . Similarly, for a proper oriented morphism F : T → X from an mdimensional stratifold with boundary we define Stokes' Theorem applied to the top stratum gives us Stokes' Theorem: Remark 4. Throughout this article, we will work with oriented morphisms and maps f : S → X and with differential forms with values in the orientation bundle of X. In the special case that X is an oriented manifold, these are ordinary forms, and an orientation of f is precisely an orientation of S. A reader not used to the more general setting might just assume the orientability and choice of orientations throughout. The passage to the general case is a mere technical point.

Differential cohomology via stratifolds
Now we define the cycles of our smooth cohomology following the recipe as for for singular cobordism [3]. The starting point is the description of k-th ordinary cohomology of X as bordism classes of continuous oriented proper maps from oriented regular stratifolds S of dimension n−k to X [9, Chapter 12]. Actually, in [9] one has to make the obvious modifications to pass from oriented manifolds to arbitrary manifolds by working with oriented maps. We call such oriented proper maps k-cycles. We denote this cohomology group SH k (X), the stratifold cohomology of X. Since every proper map is homotopic via proper maps to a morphism [9] we will always assume that f is a morphism. Let f : S → X be a k-cycle, i.e. a proper oriented morphism from a regular (n − k)-dimensional stratifold to X. Then we construct a current T (S, f ), i.e. an element in the topological dual space Ω n−k c (X; Λ X ) * of continuous linear maps from Ω n−k c (X; Λ X ) to R as follows: We have an injective map After these preparations we define a cycle for k-th smooth cohomology: where (S, f ) is as above given by a proper oriented morphism f : S → X, with S an (n − k)-dimensional oriented regular stratifold, and α ∈ Ω We define cobordisms of cycles for smooth cohomology as follows: if T is a stratifold with boundary ∂T = S and F : T → X is a proper oriented morphism, we say that (T , F ) is a zero bordism of (∂W, F | ∂T , T (T , F )), and a bordism between two cycles is a zero bordism of the difference. The only thing which is a bit special is that for the map F on a bordism T to X we require that F commutes with the collar c : ∂T × [0, ǫ) for some appropriate ǫ > 0. This allows to glue bordisms (using the collars) in a compatible way. Note also that dT (T , F) − T (∂T , F | ∂T ) = 0 (in particular is in the image of j) by (3): We denote the corresponding bordism group bŷ We call this group the smooth stratifold cohomology of X. As in [3, Definition 4.9] we define the maps The proof that these maps are well defined is literally the same as in the case, where we have smooth manifolds instead of stratifolds [3, Lemma 4.10], since the basic ingredient, Stokes' Theorem, is available.
The next aim is to construct induced maps for a smooth map g : Y → X. The basic idea is, that if (S, f, α) is a smooth cycle in X, then we can after a proper homotopy of f (which we can consider as a special case of a bordism) assume that g is transversal to f . Then, as in [9], one can consider the pull back of S giving a cycle (g * (S), F ) in Y . We denote the canonical map g * (S) → S by G. As in [3,Section 4.2.6], the orientation of f induces an orientation of F = g * f .
To extend this pull back to a smooth cycle by pulling back α, one has the same situation as in [3], i.e. one has to pull back α along g. Recall that this is only possible if W F (α)∩N (g) = ∅. Here W F (α) ⊆ T * X denotes the wave front set of the distributional form α, and N (f ) ⊆ T * X is the normal set to f . The wave front set of a distributional form α on X is a conical subset of T * X which measures the locus and the directions of the singularities of α. For a precise definition and for the properties of distributions using the wave front set needed we refer to [7,Section 8]. Compare [3, Section 4.2.6] for the notation and more details. In terms of normal sets transversality of f and g can be expressed as N (f ) ∩ N (g) = ∅ (where N (f ) is the normal set of the restriction of f to the top stratum). Hence g * α is defined if W F (α) ⊆ N (f ). In order to match this condition we use the freedom to change α by an element in the image of d * .
We observe that by definition and by Lemma 2 T (S, f ) = (f | reg S ) ! (ρ) where ρ : S reg → R is a smooth compactly supported cutoff function which is zero in a neighborhood of the singular set of S, and which is identically 1 on the support of any f * ω for ω ∈ Ω n−k c (X). Consequently, the construction of T is described entirely in the context of smooth manifolds, smooth maps and smooth forms; as in the context of [3]. Now, the arguments there, in particular [3, Lemma 4.12] literally apply in our situation to show that we can change α to a representative α ′ such that the wave front set of α satisfies W F (α ′ ) ⊆ N (f ). Then g * (α ′ ) is a well defined distribution and we can make the following definition. Definition 6. We set g * [S, f, α] := [g * (S), g * f , g * (α)], where we choose a representative such that f is transversal to g and W F (α) ⊆ N (f ).
The proof that this induced map is well defined and functorial is the same as in the case where S is a smooth manifold. Naturality of the transformations R, I and a is checked in a straigtforward way.
4 Ring structure on smooth stratifold cohomology Definition 7. We define the ×-product of classes [S, f, α] ∈ŜH k (X) and The sign (−1) kr comes from the fact that in contrast to [3] we work with orientations of the tangent bundle, whereas there normal orientations are used. This orientation convention is in agreement with that in [9].
The proof of the following fundamental properties is the same as in [3], except that for the difference of signs one has to use the arguments in [9].

Proposition 8. The product is well defined, associative, graded commutative, and natural.
Using the map induced by the diagonal ∆ : X → X × X we define the cup product.
Definition 9. For a ∈ŜH k (X) and b ∈ŜH r (X) we define By a straightforward calculation we see

Smooth stratifold cohomology as smooth extension of ordinary cohomology
We have constructed a smooth extension of the SH-homology theory as developped in [9]. However, we argue thatŜH is a smooth exension of ordinary cohomology. For this, we have to observe that the corresponding stratifold cohomology SH(X) is naturally isomorphic to ordinary cohomology H(X). The reason is that this functor fulfills the homotopy axiom (obvious) and that one has a natural Mayer-Vietoris sequence. This was proven in [9] for the case where X is oriented, but the same proof works in the non-oriented case. Now we apply [10] or [2,Section 7]. There, it is proven that a cohomology theory on smooth manifolds is naturally isomorphic to ordinary cohomology if it satisfies the homotopy axiom, the Mayer-Vietoris sequence and if the cohomology groups of a 0-dimensional manifold X are the direct product of the cohomology groups of the points in X. Moreover, it was shown in [10] that the natural isomorphism can be chosen to preserve the ring structure. Thus we can identify the multiplicative cohomology theories stratifold cohomology and ordinary cohomology. We now formulate our main theorem Theorem 11. Our constructionŜH defines a multiplicative smooth extension of ordinary cohomology with integer coefficients in the sense of Definition 1. By [11,Theorem 1.1] or [2] it follows that our theory is uniquely naturally equivalent to any other of the many models for this extension, in particular to Cheeger-Simons differential characters of [4], as described in [11]. This is actually even true as a multiplicative extension: by [11,Theorem 1.2] or [2], there is only one multiplicative smooth extension of ordinary cohomology.
Proof. Our setup is not quite identical to the one of [11], as there it is required that the kernel of R :ĤS As this is not the case, we use instead the method of proof of [2]. There, a natural transformation Φ between any two smooth extensionsĤ andĤ ′ of integral cohomology is constructed by making a universal choice. It is shown that Φ is additive and unique in even degrees because H k (pt) = 0 for k odd. The same method implies immeditately that the transformation is additive and unique in all degree except for * = 1, as H * −1 (pt) = 0 except if * = 1.
Next, the method shows that for the transformation Φ there is such that for two classes x, y of degree 1 we have However, we can modify Φ by setting Φ ′ (x) := Φ(x) − a(c) if x is of degree 1 and Φ ′ (x) = Φ(x) otherwise. Then we conclude that Φ ′ is the unique additive transformation between the two smooth extensions of integral cohomology satisfying our axioms. The methods of [2] finally show that there is at most one ring structure on a smooth extension of integral cohomology. Again, this follows because of the vanishing of H * (pt; R/Z) if * = 0, together with the consideration of distributivity for products of classes of degree zero and degree one.
Proof. This is a special case of [2, Theorem 7.12].

Integration along the fiber
Let p : X → B be the projection map of a locally trivial fiber bundle. To define "integration along the fibers" of p for a cohomology theory E, one has to choose an E-orientation for p (which might not exist). For a general cohomology theory E and a smooth extensionÊ, usually one has to choose further data in addition to an ordinary E-orientation to prescribe anÊ-orientation, compare e.g. [1, Section 3.1] or [3,Section 4.3.7].
An exception is ordinary (integral) cohomology H, as already observed in [4,5,8]. Here, an ordinary orientation determines canonically a smooth orientation and a smooth integration map. In our model of smooth cohomology using stratifolds, the definition of the integration map as well as the proof of its main properties is particularly simple. More precisely, we will prove the following theorem.
Theorem 13. Given a locally trivial smooth fiber bundle with closed ddimensional fibers which is oriented for ordinary cohomology, there is a canonical integration for smooth stratifold cohomologŷ This has the following properties 1. The smooth integration is compatible with integration of forms and of ordinary cohomology classes, i.e. the following diagrams commute:

Naturality: If
commutes, where we use on q the pullback of the orientation on p.

Functoriality: if
is a composition of two smooth oriented fiber bundles (with composed orientation), then

On a cycle
where by definition E/B α(ω) = α(p * ω) and where we equip p • f with the composed orientation.
In the remainder of this section we prove Theorem 13. Note that 5 actually is a definition ofp ! which by construction is compatible with the addition inŜH. However, we have to check that it is well defined. By compatibility with addition, for a cycle (S, f, α) representing zero, i.e. S = ∂T , f = (F : T → X)| S , α = T (T , F ), we have to check that this is mapped to zero underp ! . However, [S, p • f, p * α] is precisely the boundary of [T , p • F, T (T , F )], as by definition T is the pushdown of the fundamental class, and this is natural for composition, so 5 indeed definesp ! .
Next we prove the compatibilities of 1. Here, we have for a class of cycles x = [S, f, α] and a form ω ∈ Ω * (E): To prove naturality 2 with respect to pullback in a diagram choose (without loss of generality) the cycle x = [S, f, α] such that f is transversal to v and W F (α) ⊆ N (f ). Since p and q are submersions, the composition p•f is transversal to u. Moreover (with the notation W F y (β) = W F (β) ∩ T * y B and similar for normal sets) so that u * (p ! (x)) is defined using the cycle (S, p • f, E/B α). Then, Here we use that pullback and pushdown of distributional forms in a cartesion square are compatible (which follows from the corresponding statement for smooth forms by continuity, as the pullback is extended from smooth to distributional forms by continuity).
It remains to prove the projection formula 4. This we do in two steps. First we consider the projection id ×p : Recall finally that the cup product in smooth cohomology is defined as the pullback of the exterior product with respect to the diagonal map. We obtain p ! (p * x × y) = (−1) kd x ∪p ! (y).

Transformations between smooth cohomology
The construction of smooth cohomology via stratifolds, i.e. generalized oriented manifolds, immediately allows to define a lift of the orientation transformation from a bordism theory which is naturally equipped with an Horientation to the corresponding smooth extensions of the present article and of [3] (provided the characters are chosen appropriately).
As an example, take the canonical orientation from complex bordism to integral homology. As character on complex bordism, use this map composed with the natural map from integral cohomology to cohomology with real coefficients: In the stratifold description of integral cohomology and for X an oriented manifold, the transformation sends [E → X] ∈ M U * (X) to [E → X], where the complex oriented manifold E with proper map to X is interpreted as a stratifold with proper morphism to X. A bordism of manifolds over X is also a bordism of stratifolds, so this map is well defined; it is an easy exercise that this indeed describes the natural transformation dual to taking the fundamental class of a stable almost complex manifold.
We immediately get a smooth lift M U * (X) →ĤS * (X) by mapping theM U * (X)-class [E, f, α] to theĤS * (X)-class [E, f, α]. Obviously this is compatible with the curvature homomorphisms as well as with the passage to the underlying homology theories and the transformation ori, and with the action of differential forms.