Flowing from intersection product to cup product
Abstract.
We use a vector field flow defined through a cubulation of a closed manifold to reconcile the partially defined commutative product on geometric cochains with the standard cup product on cubical cochains, which is fully defined and commutative only up to coherent homotopies. The interplay between intersection and cup product dates back to the beginnings of homology theory, but, to our knowledge, this result is the first to give an explicit cochain level comparison between these approaches.
Key words and phrases:
Geometric cohomology, intersection product, cup product, vector field flow, manifolds with corners2020 Mathematics Subject Classification:
55N45, 57R19, 57R25Contents
1. Introduction
de Rham cohomology has long been lauded as a perfect cohomology theory by champions such as Sullivan [Sul77] and Bott [BT82]. A combination of geometric underpinning and commutativity at the cochain level make it a remarkably effective tool for many applications of rational homotopy theory. Over the integers, submanifolds and intersection in various settings provide geometrically meaningful cochains [Lip14] with a partially defined commutative product [Joy15, FMMS]. But the obstructions to commutativity witnessed by Steenrod operations show that intersection alone cannot capture the cochain quasiisomorphism type multiplicatively.
In this paper we start to marry two imperfect theories, relating multiplicative structures of, on one hand, geometric cochains defined using manifolds with corners, and, on the other, standard cubical cochains. The comparison chain map between these “analog” and “digital” presentations of ordinary cohomology of a closed manifold is defined through counting intersections of geometric cochains with a given cubulation. With the proper definitions, which we set up in detail Section 3, the map is a quasiisomorphism. In the domain of this comparison map there is a natural partially defined product given by transverse intersection, whereas in the target the product structure is induced from the Serre diagonal, a cubical analogue of the AlexanderWhitney diagonal in the simplicial setting. Both of these multiplicative structures induce the standard cup product in cohomology, but at the cochain level they are not immediately compatible. We bind them using the flow of a vector field canonically defined using the cubulation.
The basic idea of the construction is given in Figure 1. The logistic vector field is pictured in part (A) of the figure with its time flow denoted by . Part (B) of the figure illustrates the main idea of how the flow reconciles multiplications. Here and are manifolds with corners mapping to a closed manifold which we assume cubulated, focusing the picture on a single square. As explained in Section 3, such maps represent geometric cochains of , and integer coefficients can be considered if additional (co)orientation data is included. Geometric cochains that are transverse to the cubulation, as we are assuming and are, define cubical cochains and by a count of signed intersection numbers with the cubical faces. The picture shows that with modtwo coefficients evaluates to 1 on the bottom and left edges, while evaluates to 1 on the left and right edges. As explained in Section 2, because the bottom edge and right edge form an “initialterminal” pair of faces of the square, the Serre diagonal construction gives that evaluates to 1 on the square. As and do not intersect each other, their intersection product evaluates to 0 on the square, and thus disagrees with the cup product at the cochain level. Yet, for sufficiently large, and intersect while maintaining and , now yielding agreement between the intersection and cup products at the cochain level.
Our main result is that logistic flow performs such reconciliation in general. We write for the fiber product of and over , which gives rise to the partially defined product on geometric cochains, defined when and are transverse. With this notation we now present the main result of this work.
Theorem 1.1.
Let be a cubulated closed manifold and and two compact cooriented manifolds with corners over which are transverse to the cubulation. Then, for sufficiently large:

and are transverse and

and are transverse and
where is the product of the codimensions of and over .
It is classically known that the intersection and cup products are Poincaré dual at the level of homology and cohomology, but, to our knowledge, this result is the first to give an explicitly connection between these products at the cochain level.
Turning to applications, manifold cochains have primarily been developed as a parallel to, or for application in, string topology [CS99], Floer theory [Lip08], and other types of moduli questions [BJ17]. More work needs to be done for our viewpoint to connect with these fields, but as they stand, the results of this paper are applicable, for example, in using the bar construction on cochains to define knot invariants through induced maps on configuration spaces [BCSS05, SW13, BCKS17].
Since the logistic flow interpolates between commutative and noncommutative worlds, in future work we plan to connect it to cup products [Ste47, MM18] and higher derived structures [MM20a, BMMM20]. More generally, as has been done for combinatorial cochains [MS03, BF04, MM20b, MM21], our work invites the possibility of defining structures on geometric cochains and the description of cohomology operations at the cochain level [KMM20] using geometric language. We are particularly interested in building on the work of Mandell [Man01] and others to model homotopy types of manifolds via geometric cochains.
The question of relating vector field flows to finer cochain structures has also recently arisen in mathematical physics [Tho18, Tat20], but the vector fields in [Tat20] are noncontinuous. Our flow is globally smooth and thus should serve as a strong bridge between physical models, geometry, and topology.
There are two variants of Theorem 1.1 which are likely of interest but which will not be addressed in this paper. First, one can use simplices instead of cubes. Working with cubulations simplifies our treatment since the logistic flow on standard cubes is given coordinatewise by the logistic flow on the interval. But the simplicial version of Theorem 1.1 can be proven for simplicial cochains with the AlexanderWhitney product, using the results of this paper and the model of standard simplices as subsets of cubes with nonincreasing coordinates. We leave the details to the interested reader. Secondly, we conjecture that there is a version of Theorem 1.1 in which some finite subcomplex of geometric cochains maps to a version of transverse smooth singular cochains. Precise formulation of such a comparison map is one of the topics we plan to address in [FMMS], so for now we leave this idea undeveloped.
We begin the paper by reviewing in Section 2 basic material on cubical structures. We then describe geometric cochains defined using manifolds with corners, a notion that arises naturally when considering fiber products of manifolds with boundary. But for manifolds with corners, the boundary of a boundary is not empty, so one must impose a quotient at the cochain level to obtain a cochain complex. In Section 3, we review the needed parts of the this theory as given in [FMMS] and based on the original definition of Lipyanskiy [Lip14]. In Section 4, we then develop logistic vector fields, for which the analysis is thankfully simple to manage. These vector fields in a sense give a smooth extension of the cubical poset structure, the key combinatorial structure used in defining the cubical cup product. We put everything together in Section 5 to prove our main comparison theorem, stated above, which intuitively says that after sufficient time flow intersection yields a ring homomorphism from geometric cochains to cubical cochains.
Acknowledgments
The authors thank Mike Miller, for pointing us to [Lip14], and Dominic Joyce, for answering questions about his work.
2. Cubical topology
2.1. Cubical complexes
The interpolation we develop between combinatorial and smooth topology proceeds through a cubulation of a manifold – that is, a cubical complex homeomorphic to the manifold. Such a structure is less common than that of a triangulation, so we present basic definitions, in a form best suited to our applications.
For simplicial complexes, vertices can always be given a partial order that restricts to a total order on each simplex, providing a way to identify each simplex with the standard simplex. Furthermore, when two simplices meet along a common face, the induced ordering data for that face is consistent. Categorically, such data is reflected in the fact that every simplicial complex is the realization of some simplicial set. There is a parallel to this in the cubical setting, namely data required to compatibly identify each cube of a cubulation with the standard cube.
We thus begin with a formulation of cubical complexes containing such extra ordering data, as well as a description of the key features of cubical structures that will be needed for the analysis of our vector field flows in Section 4.
The standard cube is the subset of defined by
with the standard topology but with our preferred metric being the metric. Denote by . A partition of defines a face of given by
We abuse notation and use the same notation for the partition and its associated face, referring to coordinates with as free and to the others as bound. The dimension of is its number of free coordinates, and as usual the faces of dimension and are called vertices and edges, respectively. The set of vertices of is denoted by .
If , then we say that is an initial face; if , then we say that is a terminal face. Let be the union of initial faces of dimension and the union of terminal faces of dimension .
The maps are defined for and by
and any composition of these is referred to as a face inclusion map.
For all coordinates are bound – that is, . Thus is determined by the partition of into and , so we have a bijection from the set of vertices of to the power set of , sending to . The inclusion relation in the power set induces a poset structure on given explicitly by
We will freely use the identification of these posets. The smallest and largest elements in , which we denote and , are the initial and terminal vertices. Face embedding maps induce orderpreserving maps at the level of vertices.
An interval subposet of is one of the form for a pair of vertices . There is a canonical bijection between faces of and such subposets, associating to the face defined by for .
The posets play the role for cubical complexes that finite totally ordered sets play for simplicial complexes. Recall for comparison that one definition of an abstract ordered simplicial complex is as a pair , where is a poset and is a collection of subsets of , each with an induced total order, such that all singletons are in and subsets of sets in are also in . We have the following cubical analogue.
Definition 2.1.
A cubical complex is a collection of finite nonempty subsets of a poset , together with, for each , an orderpreserving bijection for some , such that:

For all , ,

For all and all the set and the following commutes
We refer to an element as a cube of , refer to as its characteristic map, and refer to as its dimension. If , we say that is a face of in . We identify elements in with the singleton subsets in , referring to them as vertices.
In analogy with the usual terminology in the simplicial setting, one could call these “ordered cubical complexes,” but we only work with these and have seen little use elsewhere for the unordered version.
Consider the category defined by the inclusion poset of a cubical complex and the subcategory of the category of topological spaces whose objects are the cubes, identified with , and whose morphisms are face inclusions. The characteristic maps of define a functor from its poset category to , and we define its geometric realization as the colimit of this functor. A cubical structure or cubulation on a space is a homeomorphism from the geometric realization of a cubical complex. We abuse notation and write simply as for any when a cubical structure is understood.
Our definition sits between cubical sets [Jar02] and cellular subsets of the cubical lattice of [KMM06], analogously to the way that abstract ordered simplicial complexes sit between simplicial sets and simplicial complexes. The geometric realization construction makes our definition and the cubical lattice definition essentially equivalent. Just as is the case for simplicial complexes, faces in cubical complexes are completely determined by their vertices.
The vector field flow we define on cubulated manifolds in Section 4 can be viewed as a smooth extension of the cubical poset structure. We refine our description of this poset structure through identifying “previous” and “next” faces in a cube.
Definition 2.2.
Let be a face of . The decomposition of is the isomorphism where and .
An alternate definition of is as the face whose initial vertex is the terminal vertex of and whose terminal vertex is , the terminal vertex of .Similarly, is the face whose terminal vertex is the initial vertex of and whose initial vertex is , the initial vertex of . See Figure 4.
The special case of decompositions in which , a vertex, merits its own consideration.
Definition 2.3.
An ordered pair of faces of is said to be reciprocal if there exists a vertex such that and . Equivalently, is reciprocal if and only if is initial and , or if and only if is terminal and .
Consider the ordered set where . For any face of , the ordered subset defines the canonical orientation of . We define the shuffle sign of , denoted by , to be if the decomposition isomorphism is orientation preserving and if not. More explicitly, if the concatenation of the ordered sets , , and represents the same orientation as , and otherwise. This sign plays a key role in our applications, since we work over the ring of integers and this sign occurs in comparing products.
2.2. Cubical cochains
We can also define an “algebraic realization” for a cubical complex in analogy to its geometric realization. Let be the usual cellular chain complex of the interval with integral coefficients. Explicitly, is generated by the vertices and , and is generated by the unique 1dimensional face, denoted in the interval subposet notation. The boundary map is .
Let , with differential defined by the graded Leibniz rule. Given a face inclusion the natural chain map is defined on basis elements by
Regarding a cubical complex as a functor to , we can compose it with the chain functor above to obtain a functor to chain complexes. The complex of cubical chains of , denoted , is defined to be the colimit of this composition. As one would expect, in each degree it is a free abelian group generated by the cubes of that dimension, and its boundary homomorphism sends the generator associated to a cube to a sum of generators associated to its codimensionone faces with appropriate signs.
The cubical cochains of (with coefficients) is the chain complex . By abuse, we use the same notation and terminology for an element in , its geometric realization in , and the corresponding basis elements in and .
We next recall the Serre diagonal. Let be defined on basis elements by
Then, let be the composite
where is the shuffle map that reorders tensor factors so that those in odd positions occur first. More explicitly, using Sweedler’s notation, if and are defined through the identity
then
(1) 
where the sign is determined by the Koszul convention.
The cup product of cochains is defined using the Serre diagonal as follows^{1}^{1}1 We follow the convention for evaluation of tensor products of cochains on tensor products of chains given by . This convention is used for defining the cup product, for example, by Munkres [Mun84, Section 60], Hatcher [Hat02, Section 3.2], and Spanier [Spa81, Section 5.6]. But it disagrees with the conventions in Dold [Dol72, Section VII.7], where there is a sign coming from the Koszul convention.:
We will use the following more explicit description of Serre’s diagonal.
Proposition 2.4.
The map satisfies
Proof.
In expression (1) each must be or and each must be or . Moreover, if then , and if then . Hence, in each summand of (1), the first and second tensor factors are reciprocal faces of . Conversely, each vertex of determines such a summand. The proposition now follows from the identification of the shuffle sign with the sign arising from applying the Leibniz rule. ∎
3. Geometric cochains
To specify a cubical cochain in a fixed degree is to give an integer for each and every cube in that dimension, which in practice can be an unwieldy amount of data. Submanifolds, which can be simple to describe in cases of interest, can encode such data through intersection.
The basic idea is classical, essentially an implementation of Poincaré duality at the chain and cochain level by using intersection with a submanifold in order to define a function on chains. We implement this idea in Definition 3.21. But there are technical obstacles to overcome in order to obtain cochain models. First, submanifolds alone do not capture homology and cohomology, as Thom famously realized and as can be seen in applications such as using Schubert varieties to represent cohomology of Grassmannians. So we generalize from submanifolds to any manifold equipped with a map to our manifold in question. These evaluate on chains through pullback, generalizing intersection. Secondly, we need manifolds with corners to define a product using fiber product, as boundaries are needed to define cohomology and corners arise immediately when taking fiber products of manifolds with boundary. Even though, for example, the collection of smooth maps from simplices constitute a cochain complex additively, taking fiber product quickly leads to more general representing objects.
While there are a number of treatments of homology and cohomology that employ manifolds and their generalizations [Whi47, BRS76, FS83, Kre10, Kah01, Zin08, Joy15], we find geometric cohomology, first developed by Lipyanskiy in the preprint [Lip14], to be the most suitable for connecting differential and combinatorial topology. In [FMMS], we have filled in details of this theory as well as equipping it with a (partially defined) multiplicative structure on cochains. We now give an overview of geometric cohomology referring to [FMMS] for a more complete exposition.
3.1. Manifolds with corners
We follow a careful development by Joyce [Joy12]. Let , and let denote projection onto the th coordinate. Define a map between open subsets of these spaces to be smooth if it can be extended to a smooth map of the ambient Euclidean space in a neighborhood of each point. We carry over the definitions of smooth charts and atlases as in the standard setting, and we choose to work with subspaces of in order to have a set of such objects. The following definitions are from [Joy12, Section 2].
Definition 3.1.
A manifold with corners, or simply a cmanifold, is a subspace of some that is a topological manifold with boundary together with an atlas of smooth local charts modeled on .
The smooth realvalued functions on a manifold with corners are those such that for each chart the composition is smooth.
A map from a manifold with corners to a manifold without boundary is smooth if the composition of with any smooth realvalued function on is a smooth realvalued function on .
The tangent bundle of a manifold with corners is the space of derivations of the ring of smooth realvalued functions.
By modeling on , our category includes manifolds () and manifolds with boundary (), as well as cubes and simplices, but not the octahedron, for example, as the cone on is not smoothly modeled by any .
Joyce extends the notion of smooth map to maps between manifolds with corners by making fairly stringent requirements on points which map to the boundary of the codomain, so that, in particular, fiber products are wellbehaved. In Joyce’s terminology, the extension of our definition of smooth maps into manifolds with corners is called weakly smooth. We will not require this more stringent definition of a smooth map between manifolds with corners, as the only time we will consider fiber products over manifolds with corners will be when those products are zerodimensional, for which we give an ad hoc treatment.
We will use boundaries of manifolds with corners, which are defined through their natural stratifications.
Definition 3.2.
A point in a manifold with corners has depth if there is a chart from an open subset of that sends the origin to . Define the cornerstrata to be the set of elements having depth .
If is a manifold with boundary, then is its interior and is its boundary. But, if is a general manifold with corners, deeper corner strata need to be incorporated in the boundary. Because of this, the boundary is naturally a cmanifold over (that is, a map from a cmanifold to ), rather than a subspace of . Again see [Joy12, Section 2] for further details.
Definition 3.3.
A local boundary component of at is a consistent choice of connected component of for any neighborhood of , with consistent meaning that .
Since this notion is local, the number of such components is determined by depth. Considering the origin in , for any , points having depth have exactly local components. For example, consists of the interiors of twodimensional faces, and any sufficiently small neighborhood of a corner intersects exactly three of these.
Definition 3.4.
Let be a manifold with corners. Define its boundary to be the space of pairs with and a local boundary component of at . Define by sending to .
The boundary is itself a manifold with corners, and the boundary map is an immersion. If is oriented, we orient by stipulating that an outward normal vector followed by an oriented basis of yields an oriented basis for . Taking boundary satisfies the Leibniz rule.
For geometric cohomology, we need coorientations rather than orientations. These are treated below and are more involved, requiring care to develop in [FMMS].
We let denote with , and we let , or simply , denote the composite of the maps sending to .
Recall the standard notion of transversality of two maps, defined locally by having the tangent space at an image point spanned by the images of tangent spaces of preimages.
Definition 3.5.
Let and be smooth maps from manifolds with corners to a manifold without boundary. We say and are transverse, denoted , if and are transverse for all . This is equivalent to requiring all pairs and be transverse in the standard sense.
Suppressing maps from the notation, define the pullback or fiber product as the subspace of with .
We will use the term pullback when we want to emphasize its map to or , while the fiber product is to be considered over . The following analysis of fiber products is standard – see for example Proposition 7.2.7 of [MROD92].
Theorem 3.6.
Let and be smooth maps from manifolds with corners to a manifold without boundary. If then the fiber product is a manifold with corners with
Moreover, the maps from the fiber product to , , and are weakly smooth.
To generalize this theorem when is also a manifold with corners requires substantial additional hypotheses in the definition of transverse smooth maps. Such a generalization is a central result in [Joy12]. We only require this case and the case of manifolds of complementary dimension, discussed below.
3.2. Geometric cohomology
Geometric cohomology is a cohomology theory for smooth manifolds defined via proper cooriented maps from manifolds with corners. It agrees with singular cohomology, but with different representatives at the cochain level it gives geometric approaches to both theory and calculations. It is thus akin to de Rham theory in being defined through smooth manifold structure rather than continuous maps. But, unlike de Rham theory, it is defined over the integers.
Geometric homology and cohomology were defined and developed in a preprint of Lipyanskiy [Lip14]. But this preprint does not develop a multiplicative structure at the cochain level. Moreover, while Lipyanskiy shows geometric homology groups are isomorphic to singular homology, he does not state or prove the corresponding fact for cohomology. We give a full treatment addressing these points and others in [FMMS], reviewing here only what we need to compare geometric and cubical cohomology.
We first define coorientations. Recall that one definition of an orientation of a bundle is an equivalence class, up to positive scalar multiplication, of an everywhere nonzero section of the top exterior power of the bundle.
Definition 3.7.
Let be a rank vector bundle. If , define to be , and if , define to be the trivial rank one bundle.
A coorientation of is an equivalence class, up to positive scalar multiplication, of an everywhere nonzero section of . We say that is coorientable if a coorientation exists.
The local triviality of the determinant line bundle of a manifold means being able to choose a consistent basis vector over sufficiently small neighborhoods. We call such a choice of basis vectors around a point, which we typically do not specify, a local orientation, and often denote the local orientation for by . We use orderedpair notation for coorientation homomorphisms, with being the coorientation that sends the local orientation at to a local orientation for .
We can equivalently define a coorientation as a choice of isomorphism , again up to positive scalar multiplication. Thus, if is coorientable and is connected, any local coorientation uniquely extends to a global coorientation. If a map is coorientable, it has exactly two coorientations, which we say are opposite, with, for example, the opposite of above being , which we also write as .
Cooriented maps compose in an immediate way, forming a category. Namely, given and coorientations and , we simply compose the latter with the pullback of the former via .
Like orientations, coorientations are “additional data.” An exception is the intrinsic coorientation of a diffeomorphism, as the differential in this special case induces a map on determinant line bundles. A key case of coorientation is the following.
Definition 3.8.
Let be an immersion with an oriented normal bundle , with local orientation denoted by . Define the normal coorientation associated to locally as , where is any choice of a local orientation of .
Conversely, if is a cooriented immersion, define the induced orientation of the normal bundle as the one whose normal coorientation agrees with the given one.
Definition 3.9.
A cmanifold over a manifold with corners is a manifold with corners with a weakly smooth, proper, cooriented map , called the reference map. Two such are equivalent if there is a diffeomorphism so that and the composite of the coorientation of with the intrinsic coorientation of yields the coorientation of . Let denote the set of proper cooriented cmanifolds over .
For an element of , we write for the codimension . Let be the free abelian group generated by , graded by the codimension , modulo the following relations:

,

, where denotes the cooriented manifold over obtained by reversing the coorientation.
We take a free abelian group and then quotient by the first relation instead of defining sum as disjoint union as in [Lip14] since we define our manifolds with corners as subspaces of a fixed universe, which complicates self addition through union. By these relations any element of is represented by a single map from a likely disconnected manifold with corners, as in particular one can find as many “copies” as one needs of any manifold with corners embedded in .
We freely and almost always abuse notation by using the domain to refer to the manifold over , not or some other symbol, letting context determine whether we are referring to the entire data or the domain. Our favorite class of cmanifolds over are submanifolds, for which this abuse is minor.
When has no boundary, we use to construct a chain complex based on these objects that will compute cohomology. To do so, we consistently coorient boundaries, using composition of coorientations.
Definition 3.10.
The standard coorientation of a boundary inclusion is the normal coorientation associated to the outwardpointing orientation of .
If is cooriented, the induced coorientation of is the composition of the standard coorientation of into with the pullback of the coorientation of .
Our cochains will be equivalence classes of cooriented cmanifolds over , under an equivalence relation we define using the following concepts, which are taken from or inspired by the definitions of [Lip14].
Definition 3.11.
Let with reference maps and . We say

and are equivalent if there is a coorientation preserving diffeomorphism such that .

is trivial if there is a diffeomorphism such that and the composite of the coorientation of with the coorientation given by is the opposite of the coorientation of .

has small rank if the differential is less than full rank at all points of .

is degenerate if it has small rank and is the union of a trivial cooriented cmanifold over and one with small rank.
Rather than small rank, Lipyanskiy uses a condition called small image. In our notation, has small image if there is an with of smaller dimension than such that . The small rank condition is thus weaker, and we find it more manageable for purposes of defining a product while still providing a theory that is isomorphic to singular cohomology [FMMS].
A key example is the interval mapping to a point, which has small rank, but its boundary does not have small rank. It is nonetheless degenerate because its boundary is trivial.
Definition 3.12.
Let be a manifold without boundary. Let denote the subgroup of generated by those equivalence classes that are either trivial or degenerate. We define the geometric cochains of , denoted , as the quotient . We denote the equivalence class of modulo by .
The definitions are arranged so that geometric cochains form a chain complex.
Proposition 3.13.
If then . Moreover, for any , .
Details can be found in [Lip14, FMMS]. Briefly, always has a action, permuting the local boundary components attached to points in . Moreover, under our coorientation conventions, the two vectors appended to the coorientation of to obtain one for over the same point in differ by a transposition, so this action is coorientation reversing. This fact about not only eventually shows that but is first needed to show that the boundary of a degenerate map is degenerate.
Definition 3.14.
Define a differential by sending to , making into a chain complex called the geometric cochain complex. We denote its homology by , the geometric cohomology of .
We focus on the case in which is a manifold without boundary primarily because the theory with boundary requires relative constructions. For example, the identity generates , but the identity is not a cocycle unless we quotient out by mappings to the boundary. A definition for manifolds with boundary, or more generally corners, would also require boundary restrictions for transversality of weakly smooth maps, as developed for example in [Joy12]. We leave such generalizations to further work.
Lipyanskiy also develops a theory of geometric chains, as opposed to cochains, using compact domains and orientations. In Section 6 of [Lip14], he shows that the homology theory based on geometric chains satisfies some of the EilenbergSteenrod axioms, which is is enough to deduce in Section 10 that geometric homology is isomorphic to singular homology. Lipyanskiy does not treat geometric cohomology in the same detail, and in particular he does not claim that it is isomorphic to singular cohomology. We prove this is true in [FMMS], but the proof requires the development of additional tools – either cubulations (or triangulations) and the intersection homomorphism as in Definition 3.21 below, or by using work of Kreck and Singhof [Kre10, KS10].
Theorem 3.15.
On the category of smooth manifolds (without boundary) and continuous maps, geometric cohomology is isomorphic to singular cohomology with integer coefficients. That is, the functors and are naturally isomorphic.
The functoriality here at the cohomology level is fully defined with respect to all continuous maps. Given an element of cohomology represented by , choose a smooth map in the homotopy class of that is transverse to and then pull back. It is shown in [FMMS] that this process gives a welldefined induced map .
There is no full functoriality at the cochain level, since an cannot be transverse to all cmanifolds over . But there is a quasiisomorphic subcomplex of consisting of cochains that are transverse to which can be pulled back. This is analogous to having only a partiallydefined fiber product, as we introduce in Section 3.4. Such “partial functoriality” of cochains will be needed for applications of the main results of this paper at the cochain level.
3.3. Cubical structures and intersections
We now bring together the two structures we have been developing, geometric cochains and cubulations. We construct a quasiisomorphism from (a subcomplex of) geometric cochains to cubical cochains, and in subsequent sections we will exhibit a vector field flow that binds these structures multiplicatively. In the de Rham setting, integration provides a relationship between differential forms and cochains. For geometric cochains the intersection homomorphism plays a similar role.
A smooth cubulation is one for which characteristic maps are smooth maps of manifolds with corners. Smooth cubulations exist for any smooth manifold, as in the following construction of [SS92]. Start with a smooth triangulation (see for example [Mun66, Theorem 10.6] for the existence of such). Consider the cell complex that is dual to its barycentric subdivision. Intersecting those dual cells with each simplex in the triangulation provides a subdivision of the simplex into cells that are linearly isomorphic to cubes. Moreover, starting with an ordered triangulation – obtained for example by taking a barycentric subdivision – such a cubical decomposition embeds cellularly into the cubical lattice of , and thus it is the geometric realization of a cubical complex.
Since cubes are oriented manifolds with corners, the cubical chain complex maps injectively to Lipyanskiy’s geometric chain complex. But in contrast with the evaluation of singular cochains on singular chains, which is purely algebraic, the natural evaluation of geometric cochains on geometric chains is defined through intersection or, more generally, pullback.
Definition 3.16.
Let be a manifold without boundary equipped with a smooth cubulation . We say that is transverse to if its reference map is transverse to each characteristic map of the cubulation.
We denote by the subcomplex of generated by maps transverse to and by the corresponding quotient by its intersection with .
We will not reference when it is clear from the context. The subsets and are welldefined chain complexes since transversality of the maps representing geometric cochains by definition includes transversality of their restrictions to all strata, in particular their boundaries.
A key technical result, whose proof is given in [FMMS], is that these transversality conditions do not change cohomology.
Theorem 3.17.
For any cubulated manifold , the inclusion is a quasiisomorphism.
We next obtain cubical cochains from elements in essentially by counting intersections. We will require reference to various components of the intersection homomorphism, so we set them aside in a series of closely related definitions.
Definition 3.18.
A signed set is a finite set with a sign function . The signed cardinality of such a set is , which we denote by .
The signed sets we count are discrete intersections – or more generally pullbacks – of manifolds with corners.
Definition 3.19.
We say that cmanifolds over a cmanifold are complementary if

their codimensions – or equivalently dimensions – add to the dimension of ,

over any with their images are disjoint, and

over they are transverse in the usual sense.
The disjointedness condition over strata is also a transversality condition, which can be viewed as a special case of a full notion of transversality over a manifold with corners as in [Joy12]. By focusing on complementary manifolds, the stringent boundary conditions for general transversality reduce to an expected disjointedness over all but the interior.
If is a cmanifold over that is transverse to a cubulation and is a cube of complementary dimension, then the intersection of the image of with is discrete. Furthermore, the pullback is finite since is proper. About any point , the reference map is locally an embedding, and thus locally has a normal bundle. As noted above, the coorientation of determines an orientation of the normal bundle, which at the intersection point can be identified with its tangent space in . Thus this orientation of the normal bundle can be compared with the standard ordering of basis vectors of at , when identified with a standard cube via its characteristic map. (This local orientation of is not immediately related to an orientation of .)
Definition 3.20.
Let be complementary, and let be cooriented while is oriented. Define to be the signed set given by the pullback, with sign function given by comparing the normal coorientation of with the orientation of . Define the intersection number to be .
Definition 3.21.
Let be a manifold without boundary with a cubulation . The intersection homomorphism
is the grading preserving linear map defined by sending to the cochain whose value on is .
The intersection of a cube with an element of that is trivial, as in Definition 3.11, will give a canceling count, and there can be no intersections with small rank reference maps. Therefore, the intersection homomorphism vanishes on , and there is an induced a map on geometric cochains. We show in [FMMS] that this is a chain map. The proof is akin to the proof that degree of a smooth map is homotopy invariant, through the classification of compact onemanifolds. Indeed, on some cube both and are counts of manifolds over , which together are boundaries of the pullback of and , a manifold.
We refer to this induced map as the intersection chain map and denote it, abusively, also by .
Theorem 3.22.
The map induced by the intersection homomorphism is a surjective quasiisomorphism.
Surjectivity is clear since we can find for any cube a small submanifold transversally passing through it at one point. The quasiisomorphism result is proven in [FMMS].
3.4. Fiber product
We now endow geometric cochains with a (graded) commutative product given by intersection of immersed submanifolds, or fiber product more generally. This product is partially defined, as it must be if it is to be commutative and induce the cup product in cohomology. The construction ends up being delicate since our cochains are themselves equivalence classes. Indeed, Lipyanskiy only discusses multiplicative structure at the level of cohomology in Section 5 of [Lip14]. Joyce’s Mcohomology [Joy15], which has more complicated representing objects, is also endowed with cochainlevel product structure, after considerable effort.
We start at the level of cmanifolds over , a manifold with no boundary. Even here, substantial care in [FMMS] is taken to define a coorientation on the fiber product of cooriented maps. We summarize the results as follows, recalling that stands for the codimension .
Theorem 3.23.
Let and be transverse cmanifolds over , a manifold without boundary, with coorientations and . There is a unique coorientation of , which depends on and , with the following properties:

Reversing either or results in reversing .

The coorientations of and , when compared by composition with the restriction of the diffeomorphism which sends to , differ by the sign .

.

Let and be immersions with oriented normal bundles, so itself is an immersion whose normal bundle is isomorphic to the direct sum of the normal bundles of and . Orient by the orientation of the normal bundle of followed by that of . Then the coorientation agrees with .
We call the coorientation the product coorientation.
Definition 3.24.
If are transverse, define to be with the product coorientation.
By Theorem 3.23, is thus a partiallydefined graded commutative ring. We next address the passage to cochains, giving a partially defined differential graded commutative algebra.
Definition 3.25.
We say that two geometric cochains are transverse if they possess representative elements in of the form and such that:

and are transverse, and

and are elements of .
With such decompositions fixed, we define the fiber product to be the geometric cochain represented by .
A delicate argument in [FMMS] gives the following.
Theorem 3.26.
The fiber product descends to a welldefined, though only partiallydefined, product on , which in turn passes to a fullydefined product on . Under the isomorphism of Theorem 3.15, the induced product on geometric cohomology agrees with cup product on singular cohomology.
4. Logistic vector field
We construct a vector field associated to a cubulation on a manifold that, in a sense, gives a smooth extension of the cubical poset structure. This vector field is a cousin of the standard vector field on a triangulated manifold whose zeros coincide with barycenters of the triangulation. They can both, for example, be used to prove the PoincaréHopf theorem, equating the signed count of zeros of a generic vector field and the Euler characteristic of a manifold.
For our applications, as in the PoincaréHopf theorem, the most significant aspect of the considered vector field is its zero locus. In the cubical context, one can naturally take products of vector fields, and in particular define a family of vector fields on cubes starting with any vector field on the interval. Such families are compatible across face structures if the vector field is zero on the boundary of the interval. We require a vector field on the interval whose only zeros are at the boundary, in which case the vector fields on cubes will only have zeros at their corners. We choose to start from arguably the simplest such vector field, which is amenable to explicit analysis and whose dynamics have been extensively studied.
Definition 4.1.
The logistic vector field over is defined by
where . We denote the time flow of along it by .
4.1. Naturality
We will exclusively consider the restriction of to the unit cube , where we have the following key compatibility, which allows us to extend this vector field to any cubulated manifold.
Lemma 4.2.
The vector fields are natural with respect to face inclusions. That is, for integers and with ,
Proof.
We compute
as claimed. ∎
Given these face compatibilities, we will typically leave off the dimension index and simply write , allowing context to determine whether we consider on or , , on one of the faces of .
Definition 4.3.
Let be a smooth cubulation of a manifold with the characteristic map of a cube . Define the logistic vector field on associated to this cubulation by, for each , applying the derivative of to the logistic vector field on .
By the fact that characteristic maps are each homeomorphisms onto their images, Lemma 4.2 implies the logistic vector field on associated to is well defined.
We can now formulate precisely a sense in which the logistic vector field gives an extension of the vertex ordering of the cubical set that cubulates . In general, the flow of a vector field at a point in a manifold defines a flowline map or in some cases . The poset structure on or can then be imposed on the flowline, well defined as it is independent of point at which the flowline is centered. In some cases these flowline posets extend to a poset structure on all of . The logistic flow is one such case, and the resulting poset structure when restricted to vertices agrees with the poset structure, which is part of the cubical structure as in Definition 2.1. Indeed, as will be immediate from our next discussion, flowlines all extend to , starting at some vertex and ending at a vertex with in the cubical ordering.
4.2. Logistic flow
We next explicitly describe the logistic flow diffeomorphism.
Lemma 4.4.
The logistic flow exists for any and any , and is explicitly given by the logistic function
In particular, if for .
The proof is to check that the given satisfy – indeed, these are standard and can be found using separation of variables – and then appeal to existence and uniqueness of single variable ordinary differential equations.
The inverse function associated to this flow is also elementary. In a single variable, if then solving the flow as expressed in Lemma 4.4 in terms of gives us
(2) 
Since
for , we have the following.
Corollary 4.5.
For every , the limits exist, and we have
By taking derivatives of the formulas in Lemma 4.4 with respect to the , treated as coordinates, we immediately identify the flow diffeomorphism on tangent spaces.
Corollary 4.6.
The Jacobian matrix representing the differential of the diffeomorphism for a fixed time is diagonal with entries
4.3. Neighborhoods
Given a face of , we will focus on two families of subsets of that are parameterized by real numbers , the lower and upper subsets of
We also need sets that are neighborhoods of these in the bound directions. Let and consist of those points whose free variables are constrained as above and whose bound variables are within of those of . For example, the first set can be described explicitly as
and the second can be described analogously. We will refer to and respectively as a lower and upper neighborhood of despite not being a subset of either. Please compare with Figure 5.
Consistent with this, the neighborhood consists of those points whose bound variables are within of those of . In particular, if is a terminal face then , and if is initial then .
Just as flow lines go between vertices of the cubulation, respecting their order, we now show that the flow takes a lower neighborhood of one face to a neighborhood of the “next” face , whose initial vertex is the terminal vertex of .
Lemma 4.7.
Let be a face of . For any and we have