Floer field philosophy

Floer field philosophy

Katrin Wehrheim Department of Mathematics, UC Berkeley CA 94720; katrin@math.berkeley.edu
Abstract.

Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension principles and provides the basic intuition for two gauge theoretic examples which conceptually frame Atiyah-Floer type conjectures in Donaldson theory as well as the relations of Heegaard Floer homology to Seiberg-Witten theory.

1. Introduction

In the 1980s the areas of low dimensional topology and symplectic geometry both saw important progress arise from the study of moduli spaces of solutions of nonlinear elliptic PDEs. In the study of smooth four-manifolds, Donaldson [14] introduced the use of ASD Yang-Mills instantons111 A smooth four manifold can be thought of as a curved -dimensional space-time. ASD (anti-self-dual) instantons in this space-time satisfy a reduction of Maxwell’s equations for the electro-magnetic potential in vacuum, which has an infinite dimensional gauge symmetry. , which were soon followed by Seiberg-Witten equations [55] – another gauge theoretic222In mathematics, “gauge theory” refers to the study of connections on principal bundles, where “gauge symmetries” arise from the pullback action by bundle isomorphisms; see e.g. [72, App.A]. PDE. In the study of symplectic manifolds, Gromov [28] introduced pseudoholomorphic curves333Symplectic manifolds can be thought of as the configuration spaces of classical mechanical systems, with the position-momentum pairing providing the symplectic structure as well as a class of almost complex structures . Pseudoholomorphic curves can then be thought of as -dimensional surfaces in a -dimensional symplectic ambient space, which can be locally described as the image of real valued functions of a complex variable that satisfy a generalized Cauchy-Riemann equation . In both subjects Floer [22, 23] then introduced a new approach to infinite dimensional Morse theory444 Morse theory captures the topological shape of a space by studying critical points of a function and flow lines of its gradient vector field. In finite dimensions it yields a complex whose homology is independent of choices (e.g. of function) and in fact equals the singular homology of the space. based on the respective PDEs. This sparked the construction of various algebraic structures – such as the Fukaya -category of a symplectic manifold [67], a Chern-Simons field theory for 3-manifolds and 4-cobordisms [16], and analogous Seiberg-Witten 3-manifold invariants [35] – from these and related PDEs, which encode significant topological information on the underlying manifolds. Chern-Simons field theory in particular comprises the Donaldson invariants of 4-manifolds, together with algebraic tools to calculate these by decomposing a closed 4-manifold into 4-manifolds whose common boundary is given by a 3-dimensional submanifold. This strategy of decomposition into simpler pieces inspired the new topic of “topological (quantum) field theory” [2, 44, 66, 91], in which the properties of such theories are described and studied.

In trying to extend the field-theoretic strategy to the decomposition of 3-manifolds along 2-dimensional submanifolds, Floer and Atiyah [3] realized a connection to symplectic geometry: A degeneration of the ASD Yang-Mills equation on a 4-manifold with 2-dimensional fibers yields the Cauchy-Riemann equation on a (singular) symplectic manifold given by the flat connections on modulo gauge symmetries. Along with this, 3-dimensional handlebodies with boundary induce Lagrangian submanifolds given by the boundary restrictions of flat connections on . Now Lagrangians555Throughout this paper, the term “Lagrangian” refers to a half-dimensional isotropic submanifold of a symplectic manifold – corresponding to fixing the integrals of motion, e.g. the momentums. are the most fundamental topological object studied in symplectic geometry. They are often studied by means of the Floer homology of pairs of Lagrangians, which arises from a complex that is generated by the intersection points and whose homology is invariant under Hamiltonian deformations of the Lagrangians. For the pair arising from the splitting of a 3-manifold into two handlebodies , these generators are naturally identified with the generators of the instanton Floer homology , given by flat connections on modulo gauge symmetries. (Indeed, restricting the latter to yields a flat connection on that extends to both and – in other words, an intersection point of with .)

These observations inspired the Atiyah-Floer conjecture

which asserts an equivalence between the differentials on the Floer complexes – arising from ASD instantons on and pseudoholomorphic maps with boundary values on , respectively. While this conjecture is not well defined due to singularities in the symplectic manifolds , and the proof of a well defined version by Dostoglou-Salamon [18] required hard adiabatic limit analysis, the underlying ideas sparked inquiry into relationships between low dimensional topology and symplectic geometry. At this point, the two fields are at least as tightly intertwined as algebraic and symplectic geometry (via mirror symmetry), most notably through the Heegaard-Floer invariants for 3- and 4-manifolds (as well as knots and links), which were discovered by Ozsvath-Szabo [56] by following the line of argument of Atiyah and Floer in the case of Seiberg-Witten theory. In both cases the concept for the construction of an invariant of 3-manifolds is the same:

  1. Split along a surface into two handlebodies with .

  2. Represent the dividing surface by a symplectic manifold and the two handlebodies by Lagrangians arising from dimensional reductions of a gauge theory which is known to yield topological invariants.

  3. Take the Lagrangian Floer homology of the pair of Lagrangians.

  4. Argue that different splittings yield isomorphic Floer homology groups – due to an isomorphism to a gauge theoretic invariant of or by direct symplectic isomorphisms for different splittings .

Floer field theory is an extension of this approach to more general decompositions of 3-manifolds, by phrasing Step 4 above as the existence of a functor between topological and symplectic categories that extends the association

It gives a conceptual explanation for Step 4 invariance proofs such as [56] which bypass a comparison to the gauge theory by directly relating the Floer homologies of Lagrangians and . Since these can arise from surfaces of different genus, the comparison between pseudoholomorphic curves in symplectic manifolds of different dimension must crucially use the fact that the Lagrangian boundary conditions encode different splittings of the same 3-manifold. Floer field theory encodes this as an isomorphism between algebraic compositions of the Lagrangians, which in turn yields isomorphic Floer homologies (a strategy that we elaborate on in §2.4 and §3.5),

Floer field theory, in particular its key isomorphism of Floer homologies [83] hinted at above, was discovered by the author and Woodward [85, 86] when attempting to formulate well defined versions of the Atiyah-Floer conjecture. While the isomorphism of Floer homologies in [83] is usually formulated in terms of strip-shrinking in a new notion of quilted Floer homology [80], it can be expressed purely in terms of Floer homologies of pairs of Lagrangians, which lie in different products of symplectic manifolds. In this language, strip shrinking then is a degeneration of the Cauchy-Riemann operator to a limit in which the curves in one factor of the product of symplectic manifolds become trivial. (For more details, see §3.5.) This relation between pseudoholomorphic curves in different symplectic manifolds then provides a purely symplectic analogue of the adiabatic limit in [18], which relates ASD instantons to pseudoholomorphic curves.

The value of Floer field theory to 3-manifold topology is mostly of philosophical nature – giving a conceptual understanding for invariance proofs and a general construction principle for 3-manifold invariants (and similarly for knots and links), which has since been applied in a variety of contexts [5, 38, 46, 60, 85, 86]. One main purpose of this paper and the content of §2 is to explain this philosophy and cast the construction principle into rigorous mathematical terms. For that purpose §2.1 gives brief expositions of the notions of categories and functors, the category , and bordism categories . After introducing the symplectic category in §2.2, the categorical structure in symplectic geometry that can be related to low dimensional topology, in §2.3 we cast the concept of Cerf decompositions (cutting manifolds into simple cobordisms) into abstract categorical terms that apply equally to bordism categories and our construction of the symplectic category. We then exploit the existence of Cerf decompositions in and together with a Yoneda functor (see Lemma 3.5.6) to formulate a general construction principle for Floer field theories. This notion of Floer field theory is defined in §2.4 as a functor that factors through . This construction is exemplified in §2.5 by naive versions of two gauge theoretic examples related to Yang-Mills-Donaldson resp. Seiberg-Witten theory in dimensions 2+1. Finally, §2.6 explains how this yields conjectural symplectic versions of the gauge theoretic 3-manifold invariants, as predicted by Atiyah and Floer.

The second purpose of this paper and content of §3 is to lay some foundations for an extension of Floer field theories to dimension 4. Our goal here is to provide a rigorous exposition of the algebraic language in which this extension principle can be formulated – at a level of sophistication that is easily accessible to geometers while sufficient for applications. Thus we review in detail the notions of 2-categories and bicategories in §3.1, including the 2-category of (categories, functors, natural transformations), explicitly construct a bordism bicategory in §3.2, and summarize notions and Yoneda constructions of 2-functors between these higher categories in §3.3. Moreover, §3.5 outlines the construction of symplectic 2-categories, based on abstract categorical notions of adjoints and quilt diagrams that we develop in §3.4. The latter transfers notions of adjunction and spherical string diagrams from monoidal categories into settings without natural monoidal structure. This provides sufficient language to at least advertise an extension principle which we further discuss in [79]:

Any Floer field theory which satisfies a quilted naturality axiom has a natural extension to a 2-functor .

This says in particular that any 3-manifold invariant which is constructed along the lines of the Atiyah-Floer conjecture naturally induces a 4-manifold invariant. While it does seem surprising, such a result could be motivated from the point of view of gauge theory, since the Atiyah-Floer conjecture and Heegaard-Floer theory were inspired by dimensional reductions of 3+1 field theories . It also can be viewed as a pedestrian version of the cobordism hypothesis [44]666 Lurie’s constructions involve the canonical extension of a functor to . However, this requires an extension of the field theory to dimensions 1 and 0 (which we do not even have ideas for) as well as a monoidal structure on the target category (which is lacking at present because the gauge theoretic functors are well defined only on the connected bordism category). On the other hand, we have other categorical structures at our disposal, which we formalize in §3.4 as the notion of a quilted 2-category (akin to a spherical 2-category as described in [45]). In that language, the diagram of a Morse 2-function as in [26] expresses a 4-manifold as a quilt diagram in the bordism bicategory . Now the key idea for [79] is that a functor translates the diagram of a 4-manifold into a quilt diagram in the symplectic 2-category in §3.5, where it is reinterpreted in terms of pseudoholomorphic curves., saying that a functor (where the stands for compatibility with diffeomorphisms of 3-manifolds) has a canonical extension .

Finally, the extensions of Floer field theory to dimension 4 are again expected to be isomorphic to the associated gauge theoretic 4-manifold invariants, in a way that is compatible with decomposition into 3- and 2-manifolds. We phrase these expectations in §3.6 as quilted Atiyah-Floer conjectures, which identify field theories . The last section §3.6 also demonstrates the construction principle for 2-categories via associating elliptic PDEs to quilt diagrams in several more gauge theoretic examples, which provide not only the proper context for stating all the generalized Atiyah-Floer conjectures, but also yield conceptually clear contexts for the various approaches to their proofs.

While this 2+1 field theoretic circle of ideas has been and used in various publications, its rigorous abstract formulation in terms of a notion of “category with Cerf decompositions” is new to the best of the author’s knowledge. Similarly, the notions of bordism bicategories, the symplectic 2-category, generalized string diagrams, and field theoretic proofs of Floer homology isomorphisms have been known and (at least implicitly) used in similar contexts, but are here cast into a new concept of “quilted bicategories” which will be central to the extension principle – both of which seem significantly beyond the known circle of ideas. Finally, note that Floer field theory should not be confused with the symplectic field theory (SFT) introduced by [21], in which another symplectic category – given by contact-type manifolds and symplectic cobordisms – is the domain, not the target of a functor.

We end this introduction by a more detailed explanation of the notion of an “invariant” as it applies to the study of topological or smooth compact manifolds, and a very brief introduction to the resulting classification of manifolds.

1.1. A brief introduction to invariants of manifolds

In order to classify manifolds of a fixed dimension up to diffeomorphism, one would ideally like to have a complete invariant . Here is the category of -manifolds and diffeomorphisms between them (see Example 2.1.4), and is a category such as with trivial morphisms or the category of groups and homomorphisms. Such is an invariant if it is a functor (see Definition 2.1.3), since this guarantees that diffeomorphic manifolds are mapped to isomorphic objects of (e.g. the same integer or isomorphic groups). In other words, functoriality guarantees that induces a well defined map from diffeomorphism classes of manifolds to e.g.  or isomorphism classes of groups. Such an invariant lets us distinguish manifolds: If are not isomorphic (i.e. ) then and cannot be diffeomorphic. Moreover, an invariant is called “complete” if an isomorphism implies the existence of a diffeomorphism , i.e. .

Simple examples of invariants – when restricting to compact oriented manifolds – are the homology groups for fixed or their rank, i.e. the Betti numbers . These are in fact topological – rather than smooth – invariants since homeomorphic – rather than just diffeomorphic – manifolds have isomorphic homology groups. The 0-th Betti number is complete for since it determines the number of connected components, and there is only one compact, connected manifold of dimension 0 (the point) or 1 (the circle). The first more nontrivial complete invariant – now also restricting to connected manifolds – is the first Betti number , since compact, connected, oriented 2-manifolds are determined by their genus .

The fundamental group is not strictly well defined since it requires the choice of a base point and thus is a functor on the category of manifolds with a marked point. However, for connected manifolds it still induces a well defined map from manifolds modulo diffeomorphism to groups modulo isomorphism, since change of base point induces an isomorphism of fundamental groups. Viewing this as an invariant, it is complete for . In dimension , completeness would mean that the isomorphism type of the fundamental group of a (compact, connected) 3-manifold determines the 3-manifold up to diffeomorphism. This is true in the case of the trivial fundamental group: By the Poincaré conjecture, any simply connected 3-manifold “is the 3-sphere”, i.e. is diffeomorphic to . It is also true for a large class (irreducible, non-spherical) of 3-manifolds, but there are plenty of groups that can be represented by many non-diffeomorphic 3-manifolds, e.g. lens spaces and connected sums with them (see [30, 1] for surveys). Thus is a useful but incomplete invariant of closed, connected 3-manifolds. In dimension however, the classification question should be posed for fixed since on the one hand any finitely presented group appears as the fundamental group of a closed, connected -manifold, and on the other hand the classification of finitely presented groups is a wide open problem itself.777 As a matter of curiosity: The group isomorphism problem – determining whether different finite group presentations define isomorphic groups – is undecidable, i.e. cannot be solved for all general presentations by an algorithm; see e.g. [33].

Moreover, while in dimension , the classifications up to homeomorphism and up to diffeomorphism coincide (i.e. topological -manifolds can be equipped with a unique smooth structure), these differ in dimensions . In dimension , both classifications can be undertaken with the help of surgery theory introduced by Milnor [53]. In dimension 4, the classification of smooth 4-manifolds differs drastically from that of topological manifolds (see [65] for a survey). Here gauge theory – starting with the work of Donaldson, and continuing with Seiberg-Witten theory – is the main source of invariants which can differentiate between different smooth structures on the same topological manifold. In particular, Donaldson’s first results using ASD Yang-Mills instantons [14] showed that a large number of topological manifolds (those with non-diagonalizable definite intersection form ) in fact do not support any smooth structure.

2. Floer field theory

2.1. Categories and functors

Definition 2.1.1.

A category888 Throughout, all categories are meant to be small, i.e. consist of sets of objects and morphisms. However, we will usually neglect to specify constructions in sufficient detail – e.g. require manifolds to be submanifolds of some – in order to obtain sets. consists of

  • a set of objects,

  • for each pair a set of morphisms ,

  • for each triple a composition map

such that

  • composition is associative, i.e. we have for any triple of composable morphisms ,

  • composition has identities, i.e. for each there exists a unique999 Note that uniqueness follows immediately from the defining properties: If is another identity morphism then we have . morphism such that and hold for any and .

The very first example of a category consists of objects that are sets (possibly with extra structure such as a linear structure, metric, or smooth manifold structure), morphisms that are maps (preserving the extra structure), composition given by composition of maps, and identities given by the identity maps. The following bordism categories contain more general morphisms, which are more rigorously constructed in Remark 3.2.2.

Figure 1. The bordism category .
Example 2.1.2.

The bordism category in dimension is roughly defined as follows; see Figure 1 for illustration.

  • Objects are the closed, oriented, -dimensional manifolds .

  • Morphisms in are the compact, oriented, -dimensional cobordisms with identification of the boundary , modulo diffeomorphisms relative to the boundary.

  • Composition of morphisms and is given by gluing along the common boundary.

Here one needs to be careful to include the choice of boundary identifications in the notion of morphism. Thus a diffeomorphism can be cast as a morphism

(2.1.1)

given by the cobordism with boundary identifications and , as illustrated in Figure 2. In that sense, the identity morphisms are given by the identity maps .

Equipping the composed morphism with a smooth structure moreover requires a choice of tubular neighbourhoods of in the gluing operation. The good news is that gluing with respect to different choices yields diffeomorphic results, so that composition is well defined. The interesting news is that this ambiguity in the composition precludes the extension to a 2-category; see Example 3.2.3.

The notion of categories becomes most useful in the notion of a functor relating two categories, since preservation of various structures (composition and identities) can be expressed efficiently as “functoriality”.

Definition 2.1.3.

A functor between two categories consists of

  • a map between the sets of objects,

  • for each pair a map ,

that are compatible with identities and composition, i.e.

For example, the inclusion of diffeomorphisms into the bordism category in Example 2.1.2 can be phrased as a functor as follows.

Example 2.1.4.

Let be the category consisting of the same objects as , morphisms given by diffeomorphisms, and composition given by composition of maps. Then there is a functor given by

  • the identity map between the sets of objects,

  • for each pair of diffeomorphic -manifolds the map that associates to a diffeomorphism the cobordism defined in (2.1.1).

A more algebraic example of a category is given by categories and functors.

Example 2.1.5.

The category of categories consists of

  • objects given by categories ,

  • morphisms in given by functors ,

  • composition of morphisms given by composition of functors – i.e. composition of the maps on both object and morphism level.

2.2. The symplectic category

The vision of Alan Weinstein [87] was to construct a symplectic category along the following lines. (See [11, 51] for introductions to symplectic topology.)

  • Objects are the symplectic manifolds .

  • Morphisms are the Lagrangian submanifolds101010 Other terms for a Lagrangian, viewed as a morphism , are “Lagrangian relation” or “Lagrangian correspondence”, but we will largely avoid such distinctions in this paper. , where we denote by the same manifold with reversed symplectic structure.

  • Composition of morphisms and is defined by the geometric composition (where denotes the diagonal)

This notion includes symplectomorphisms , as morphisms given by their graph . Also, geometric composition is defined exactly so as to generalize the composition of maps. That is, we have . On the other hand, this more generalized notion allows one to view pretty much all constructions in symplectic topology as morphisms – for example, symplectic reduction from to is described by a Lagrangian -sphere ; see [29, 87, 80] for details and more examples.

Unfortunately, geometric composition generally – even after allowing for perturbations (e.g. isotopy through Lagrangians) – at best yields immersed or multiply covered Lagrangians.111111 Even the question of finding a Lagrangian with embedded composition was open until the recent construction of a new Lagrangian embedding in [12]. However, Floer homology121212Floer homology is a central tool in symplectic topology introduced by Floer [23] in the 1980s, inspired by Gromov [28] and Witten [90]. It has been extended to a wealth of algebraic structures such as Fukaya categories; see e.g. [67]. It can be thought of as the Morse homology of a symplectic action functional on the space of paths connecting two Lagrangians, and recasts the ill posed gradient flow ODE as a Cauchy-Riemann PDE (whose solutions are pseudoholomorphic curves). is at most expected to be invariant under embedded geometric composition, i.e. when the intersection in

(2.2.1)

is transverse, and the projection is an embedding. In the linear case – for symplectic vector spaces and linear Lagrangian subspaces – this issue was resolved in [29] by observing that linear composition, even if not transverse, always yields another Lagrangian subspace. In higher generality, and compatible with Floer homology, a symplectic category was constructed in [84] by the following general algebraic completion construction for a partially defined composition.

Definition 2.2.1.

The extended symplectic category is defined as follows.

  • Objects are the symplectic manifolds .

  • Simple morphisms are the Lagrangian submanifolds .

  • General morphisms are the composable chains of simple morphisms between symplectic manifolds .

  • Composition of morphisms and is given by algebraic concatenation .

For this to form a strict category, we include trivial chains of length as identity morphisms.

While this is a well defined category, its composition notion is not related to geometric composition yet. However, the following quotient construction ensures that composition is given by geometric composition when the result is embedded.

Definition 2.2.2.

The symplectic category is defined as follows.

  • Objects are the symplectic manifolds .

  • Morphisms are the equivalence classes in .

  • Composition is induced by the composition in .

Here the composition-compatible equivalence relation on the morphism spaces of is obtained as follows.

  • The subset of geometric composition moves consists of all pairs and for which the geometric composition is embedded as in (2.2.1).

  • The equivalence relation on is defined by if there is a finite sequence of moves in which each move replaces one subchain of simple morphisms by another,

    according to a geometric composition move resp. .

The result of this quotient construction is that the composition of morphisms is given by geometric composition if the latter is embedded. We will later recast this construction in terms of an extension of the symplectic category to a 2-category in which the equivalence relation is obtained from 2-isomorphisms; see Example 3.1.6 and §3.5.

Remark 2.2.3.

The present equivalence relation does not identify a Lagrangian with its image under a Hamiltonian symplectomorphism . Indeed, any morphism in induces a (Lagrangian where immersed) subset of by complete geometric composition, and this subset is invariant under geometric composition moves. However, such equivalences under Hamiltonian deformation can also be cast as 2-isomorphisms; see Example 3.5.1.

2.3. Categories with Cerf decompositions

The basic idea of Cerf decompositions is to decompose a -manifold into simpler pieces by cutting at regular level sets of a Morse function as illustrated in Figure 4 below. By viewing as a cobordism between empty sets, i.e. as a morphism in , this can be seen as a factorization in . Here the Morse function and regular levels can be chosen such that each piece contains either none or one critical point, and thus is either a cylindrical cobordism – diffeomorphic to the product cobordism as in (2.1.1) – or a handle attachment as in the following remark. These “simple cobordisms” are illustrated in figures 2 and 3.

Figure 2. A cylindrical cobordism supports a Morse function without critical points, whose gradient flow induces a diffeomorphism to the product cobordism with natural identification and boundary identification arising from the flow.
Remark 2.3.1.

A k-handle attachment of index is a -dimensional cobordism, which is obtained by attaching to a cylinder a handle along an attaching cycle , as illustrated in figure 3. Here denotes a -dimensional ball with boundary .

Figure 3. Handle attachments in dimension are “simple cobordisms” which support a Morse function with a single critical point of index . The attaching cycles are given by intersection of the unstable manifold (in red) with the boundary. Index and handle attachments are adjoint via orientation reversal and only appear between the empty set and sphere . Index and handle attachments are adjoint via orientation reversal (which interchanges unstable and stable manifolds) and appear between surfaces of adjacent genus .

By reversing the orientation and boundary identifications of any -handle attachment from to , we obtain a cobordism from to . This reversed cobordism is also a -handle attachment for an attaching cycle . It moreover is the adjoint of in the sense of Remark 2.4.3 and will become useful in the formulation of Cerf moves below.

Specifying to dimension and the connected bordism category, it will suffice to consider -handle attachments (and their adjoints) with attaching circles that are homologically nontrivial and thus do not disconnect the surface. More precisely, any attaching circle in a closed surface determines a -handle attachment as follows: Replacing an annulus neighbourhood of by two disks specifies a lower genus surface together with a diffeomorphism . Given this construction, the -handle attaching cobordism from to is unique up to diffeomorphism fixing the boundary.

More detailed introductions to Cerf theory can be found in e.g. [13, 27, 54]. Here we concentrate on the algebraic structure that it equips the bordism categories with. To describe this structure, we may think of Cerf decompositions as a prime decomposition of -manifolds, and more generally of -cobordisms: A decomposition into simple cobordisms (cylindrical cobordisms and handle attachments) always exists and simple cobordisms have no further simplifying decomposition. And while these Cerf decompositions are not unique, any two choices of decomposition are related via just a few moves, some of which are shown in Figure 4. These moves reflect changes in the Morse function (critical point cancellations and critical point switches), cutting levels (cylinder cancellation), and the ways in which pieces are glued together (diffeomorphism equivalences which in particular encode handle slides). All of these Cerf moves are local in the sense131313 While a diffeomorphism equivalence is not local, it decomposes into a sequence of local moves. that they replace only one or two consecutive cobordisms by one or two consecutive cobordisms with the same composition. That is, the moves are of one of three forms:

In the following, we will cast this notion – decompositions into simple pieces that are unique up to a set of moves – into more formal terms. For that purpose we denote the union of all morphisms of a category by , and we denote all relations between composable chains141414 Throughout, we will use the term “composable chain” to denote ordered tuples of morphisms, in which each consecutive pair is composable, so that the entire tuple – by associativity of composition – has a well defined composition. of morphisms by

Definition 2.3.2.

A category with Cerf decompositions is a category together with

  • a subset of simple morphisms,

  • a subset of local Cerf moves, which is symmetric (under exchanging the factors) and consists of pairs of composable chains of simple morphisms , whose compositions are equal,

such that

  • the simple morphisms generate all morphisms, i.e. for any there exist such that ,

  • the presentation in terms of simple morphisms is unique up to Cerf moves, i.e. any two presentations of the same morphism in terms of and are related by a finite sequence151515 Throughout, we will use the term “sequence” to denote a finite totally ordered set. of identities

    in which each equality replaces one subchain of simple morphisms by another,

    according to a local Cerf move .

The bordism categories are the motivating example of categories with Cerf decompositions, with and given by the simple cobordisms and Cerf moves as discussed above (for a more detailed exposition see [27]). However, in the examples arising from gauge theory, we consider the -dimensional connected bordism category, the case of the following general notion for .161616 We restrict to dimension when discussing connected bordisms since the handle attachments in dimension are morphisms between generally disconnected -manifolds, so that does not have useful connected Cerf decompositions.

Example 2.3.3.

The connected bordism category is defined as follows.

  • Objects are the closed, connected, oriented -dimensional manifolds.

  • Morphisms are the compact, connected, oriented -dimensional cobordisms with identification of the boundary, and modulo diffeomorphisms as in .

  • Composition is by gluing via boundary identifications as in .

If we allow as object, then closed, connected, oriented -manifolds are contained in this category as morphisms from to .

In this language, the Cerf decomposition theorem for -manifolds – in the connected case proven in [26] and reviewed in [27] – can be stated as in the following theorem, and is illustrated in Figure 4 and further explained in Remark 2.3.5. Here, in strict categorical language, a -cobordism from to is an equivalence class of -cobordisms and embeddings modulo diffeomorphisms relative to the boundary identifications . However, the decomposition and boundary identifications are actually induced by a decomposition of representatives, thus we drop the brackets and embeddings – see [27] and §3.2 for more deliberations on this. Moreover, we may again generalize to dimension .

Figure 4. Cerf decompositions of a 3-cobordism and Cerf moves between them.
Theorem 2.3.4.

is a category with Cerf decompositions as follows.

  • The set of simple morphisms consists of

    • cylindrical cobordisms for diffeomorphisms as in (2.1.1),

    • k-handle attachments for as in Remark 2.3.1.

  • The Cerf moves are the following and their transpositions:

    • Cylinder cancellations for all composable pairs of diffeomorphisms .

    • Cylinder cancellations resp.  in which is the same cobordism as (up to diffeomorphism), but with incoming resp. outgoing boundary inclusion pre- resp. post-composed with a diffeomorphism .

    • Critical point cancellations occur for attaching cycles with transverse intersection in a single point; these give rise to a pair of cobordisms , whose composition is a cylindrical cobordism representing a diffeomorphism .

    • Critical point switches and occur for disjoint attaching cycles ; these give rise to a pair of cobordisms171717 See Remark 2.5.1 for more details on the notation used here. , whose composition is the same as that of the pair , .

Remark 2.3.5.

For the objects of – closed, connected, oriented surfaces – can be classified up to diffeomorphism by their genus. Moreover, the simple morphisms can be further specified:

  • Cylindrical cobordisms represent diffeomorphisms between surfaces of the same genus as in (2.1.1).

  • 2-Handle attachments specified by a homologically nontrivial circle are simple morphisms181818 More precisely, is obtained by attaching to the cylindrical cobordism a 2-handle along a thickening of the attaching circle. from a surface of genus to a surface of genus .

  • 1-Handle attachments are 2-handle attachments with reversed orientation, i.e. the simple morphisms from a surface of genus to a surface of genus .

The structural similarities between the symplectic and bordism categories can now be phrased in terms of abstract Cerf decompositions.

Lemma 2.3.6.

The symplectic category from Definition 2.2.2 is a category with Cerf decompositions as follows:

  • The set of simple morphisms consists of the equivalence classes of Lagrangian submanifolds .

  • The set of local Cerf moves consists of the relations

    for embedded geometric compositions as in (2.2.1).

Proof.

To check that the simple morphisms generate all morphisms, consider a general morphism and pick a representative , given by a composable chain of Lagrangian submanifolds from to . The definition of composition in yields the identity

Since each is a simple morphism, this is the required decomposition of into simple morphisms. To show that these decompositions are unique up to the given Cerf moves, note that an equality

in means by definition that the corresponding morphisms in are equivalent

under the equivalence relation given in Definition 2.2.2. Recall that this relation is generated by the geometric composition moves , so that there is a sequence of moves from to in which adjacent pairs are replaced by their embedded geometric composition. Our definition of by moves on equivalence classes encoded by translates this into a sequence of Cerf moves from to . ∎

2.4. Construction principle for Floer field theories

The algebraic background of Floer field theory is the following construction principle for functors between categories with Cerf decompositions.

Lemma 2.4.1.

Let be two categories with Cerf decompositions and a Cerf-compatible partial functor consisting of

  • a map ,

  • a map which induces a map given by

Then has a unique extension to a functor which restricts to on and .

Proof.

Compatibility of with composition requires its value on a general morphism to be for any Cerf decomposition into simple morphisms . The induced map guarantees that this definition of is independent of the choice of decomposition, thus yields a well defined map . Moreover, this map is compatible with composition by construction. Thus a well defined functor is uniquely determined by . ∎

The next Lemma specializes this abstract construction principles to and and is illustrated in Figure 5. It can be read in two ways: In the strictly categorical sense, a partial functor should assign to a class of simple cobordisms modulo diffeomorphisms relative to the boundary identifications a class