Katrin Wehrheim and Chris T. Woodward- Orientations for Pseudoholomorphic Quilts

8/3/2019 Katrin Wehrheim and Chris T. Woodward- Orientations for Pseudoholomorphic Quilts

1/44

ORIENTATIONS FOR PSEUDOHOLOMORPHIC QUILTS

KATRIN WEHRHEIM AND CHRIS T. WOODWARD

Abstract. We construct coherent orientations on moduli spaces of quilted pseudoholo-morphic surfaces and determine the effect of various gluing operations on the orientations.We also investigate the behavior of the orientations under composition of Lagrangian cor-respondences.

Preliminary version 2/1/2007.

Contents

1. Introduction 12. Orientations for Cauchy-Riemann operators 2

3. First relative non-Abelian cohomology 134. Orientations for Cauchy-Riemann operators, continued 195. Orientations for pseudoholomorphic surfaces 316. Orientations for diagonal insertions 367. Orientations for compositions of Lagrangian correspondences 42References 44

1. Introduction

This paper constructs coherent orientations on moduli spaces of pseudoholomorphicquilts, introduced in our earlier paper [14]. For pseudoholomorphic disks the construction isoutlined in Fukaya-Oh-Ohta-Ono [3]. Some of the details are described in Ekholm-Etnyre-Sullivan [2]. A slightly different construction is given in Seidels book draft [9]. Most of theproofs in this paper are slight modifications of proofs in one of these three sources, and sothe paper should be considered largely expository. However, even for pseudoholomorphicdisks some of the material is new; for instance, we treat the behavior of orientations undergluing a disk to itself by a pair of points on the boundary, as well as the case of multipleoutgoing ends. We also give a more general treatment of relative spin structures, whichavoids the triangulations used in [3].

The construction of coherent orientations on these moduli spaces allows the definition ofFloer homology and the relative invariants associated to strip-like ends with integer coeffi-cients. In particular, the Donaldson-Fukaya category associated to a monotone symplectic

manifold becomes a category whose Hom spaces areZ

-modules (if one restricts the objectsto those with minimal Maslov number at least three) or, more generally, objects in thederived category of matrix factorizations over the integers. In separate papers [15], [6], weuse this material to (i) construct a long exact sequence in Floer homology for fibered Dehn

1


2/44

2 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

twists, extending results of Seidel [10] for Lagrangian spheres from Z2 to Z, and (ii) extendthe construction of Fukayas A category associated to a monotone symplectic manifold toinclude generalized Lagrangian submanifolds as objects.

The first part of the construction, covered in Section 2, is a purely linear constructionthat associates to a Cauchy-Riemann operator an orientation of its determinant line. Thispart involves various choices, analogous to the choice of orientations on the stable manifolds

used in the construction of Morse homology over the integers. To pass to the non-linearcase in Section 4, one needs a topological structure (the relative spin structures of [3]) onthe tangent bundle to the Lagrangians that guarantees that the index bundle is orientable.This is a special case of the material in Section 3 on non-Abelian cohomology relativeto a smooth map and quotient by a central subgroup. Sections 6 and 7 of the paperinvestigate the behavior of the orientations under the operations of inserting a diagonaland composition of Lagrangian correspondences.

2. Orientations for Cauchy-Riemann operators

2.1. Determinant lines. Let V, W be real Banach spaces, and Fred(V, W) the space ofFredholm operators D : V W, that is, operators with closed range and finite dimensional

kernel and cokernel. The index of a Fredholm operator D : V W is the integerInd(D) = dim(ker(D)) dim(coker(D)).

The determinant line of a Fredholm operator D : V W is

det(D) = max(coker(D)) max(ker(D)).

If D1 : V1 W1, D2 : V2 W2 are Fredholm operators then we have an equality of indices

Ind(D1 D2) = Ind(D1) + Ind(D2).

and a canonical isomorphism of determinant lines

(1) det(D1 D2) det(D1) det(D2).

Explicitly ifvk,i is a basis for ker(Dk) and wk,i a basis for coker(Dk) then the isomorphism

is defined byi

w2,i i

w1,i

i

v1,i i

v2,i

(1)dim(coker(D2)) Ind(D1)

i

w1,i i

v1,i

i

w2,i i

v2,i

.

The isomorphism (1) is associative and graded commutative in the following sense: Thecomposition

(2) det(D2) det(D1) det(D2 D1) det(D1 D2) det(D1) det(D2),

where the middle map is induced by exchange of summands, agrees with the map inducedby exchange of factors by a sign (1)Ind(D1) Ind(D2).


3/44

ORIENTATIONS FOR PSEUDOHOLOMORPHIC QUILTS 3

If D : V W is a linear operator on finite dimensional spaces, then there is a canonicalisomorphism to the determinant of the trivial operator from V to W,

(3) tD : det(D) det(0) = max(W) max(V).

To define this explicitly, choose bases e1, . . . , en for V and f1, . . . , f m for W so that D(ej) =fj for j = 1, . . . , k and D(ej) = 0 for j = k + 1, . . . , n. Let f

1 , . . . , f

m be the dual basis for

W, then we define

tD((fn . . . f

k+1) (ek+1 . . . em)) := (f

n . . . f

1 ) (e1 . . . em).

Note that tD is independent of the choice of bases ei, fj .The construction of determinant lines works in families: For a topological space X con-

sider Fredholm morphisms D : V W of Banach vector bundles V X, W X. Thedeterminant line bundle ofD is a line bundle over X. In particular, any homotopy of Fred-holm operators Dt, t [0, 1] induces a determinant line bundle over X = [0, 1]. Trivializingthe line bundle induces an isomorphism of determinant lines det(D0) det(D1). In partic-ular, taking X = Fred(V, W) and trivial bundles with fibers V, W, the universal Fredholmsection Fred(V, W) V Fred(V, W) W, (D, v) (D,Dv) determines a determinantline bundle over Fred(V, W).

Remark 2.1.1. For V separable and infinite dimensional, Fred(V, V) is the classifying spacefor real K-theory: KO(X) = [X, Fred(V, V)]. Let Pic(X) denote the Picard group ofisomorphism classes of real line bundles on X. Pulling back det(V, V) under the classifyingmap defines a homomorphism det : KO(X) Pic(X).

2.2. Orientations for Fredholm operators. Let V be a finite dimensional vector space,and max(V) its top exterior power. An orientation for V is a component of maxV \ {0},that is, a non-vanishing element of maxV up to homotopy. An oriented vector space is avector space equipped with an orientation. Given an oriented vector space V, we say thata basis e1, . . . , en of V is oriented if e1 . . . en defines the orientation on V. A linearisomorphism T : V W induces a map on orientations. If V and W are oriented, we saythat T acts by 1 on the orientations if T is orientation preserving resp. reversing.

An orientation for V induces an orientation for the dual V

. Explicitly, if e1, . . . , en isan oriented basis for V and e1 , . . . , en the dual basis, we give V

the orientation defined by

(4) en . . . e1

max(V).

Note the reverse order. If we identify V with V by an inner product, then the orientationon V differs from the pull-back orientation on V by a factor (1)dim(V)(dim(V)1)/2. Thisis opposite the convention of [2].

Orientations on finite dimensional vector spaces V, W induce an orientation on the directsum V W as follows. Given oriented bases e1, . . . , en resp. f1, . . . , f m for V resp. W defineon the sum V W the orientation given by e1 . . . en f1 . . . fm

max(V W).The isomorphism V W W V given by transposition acts on orientations by a sign(1)dim(V)dim(W).

An orientation of a Fredholm operator D : V W is an orientation of its determinantline. For finite-dimensional V, W, orientations on V and W induce an orientation on det(0),and by (3) on det(D). By convention (4) this definition is compatible with the canonicalorientation on det(Id) = R for the identity operator if V = W.


4/44


Remark 2.2.1. Let Fred+(V, W) denote the space of Fredholm operators D : V W,equipped with orientations of their determinant bundles det(D). Thus Fred+(V, W) isa double cover of Fred(V, W), so that the pull-back of the determinant line bundle toFred+(V, W) is orientable.

Let V be infinite-dimensional and separable. The oriented real K-theory of a space X isthe set of homotopy classes of maps KS O(X) := [X, Fred+(V, V)]. Let Pic+(X) be the set

of isomorphism classes of oriented real line bundles, equipped with group structure given bytensor product. By (2), the direct sum operation gives KS O(X) the structure of a gradedAbelian group, so that det : KS O(X) Pic+(X) is a group homomorphism.

2.3. Cauchy-Riemann operators. Cauchy-Riemann operators on surfaces with bound-ary. Let S be a compact, holomorphic surface with boundary, E a complex vector bundleover S with a maximally totally real subbundle F E|S; that is

F iF = {0}, rankR(F) = rankR(E)/2 = rankC(E).

Let 0(E, F) denote the space of sections of E with boundary values in F. An operatorD : 0(E, F) 0,1(E) is a Cauchy-Riemann operator if it is complex linear and satisfies

the Leibniz rule D(f ) = f D() + (f)()

for all f C(S,C), 0(E, F). The set of all Cauchy-Riemann operators is anaffine space modelled on 0,1(S, End(E)). A real Cauchy-Riemann operator is the sum of aCauchy-Riemann operator with a zeroth order term taking values in EndR(E). These forman affine space modelled on 0,1(S) R EndR(E); in particular this space is contractible.

Riemann-Roch, Serre duality, and Kodaira vanishing all have generalizations to Cauchy-Riemann operators on surfaces with boundary. Let DE,F denote a real Cauchy-Riemannoperator acting on sections of E with boundary values in F. Riemann-Roch for surfaceswith boundary [7, Appendix] gives

(5) Ind(DE,F) = rankR(F)(S) + I(E, F),

where (S) is the Euler characteristic of S and I(E, F) is the Maslov index of the pair(E, F). The cokernel of DE,F can be identified with the kernel of the adjoint D

E,F. The

operator DE,F is a real Cauchy-Riemann operator acting on sections of (E (T S)) =

Hom(E T S,C) with boundary values in the subbundle (F T(S))ann, the real sub-bundle of E (T S) whose evaluations on F T(S) vanish. The index identities

I(TS ,T(S)) = 2(S), I(E, Fann) = I(E, F)

show consistency with Ind(DE,F) = Ind(DE,F). Indeed,

Ind(DE,F) = rankR((F T(S))ann)(S) + I(E (T S), (F T(S))ann)(6)

= rankR(F)(S) rankC(E) 2(S) I(E, F).(7)

On a disk, this duality changes a Maslov index I(E, F) into a Maslov index 2rankR

(F) I(E, F), while on the annulus the pair (TS ,T(S)) is trivial so there is no shift.Starting from a surface with boundary S let S be the surface without boundary obtained

by gluing together two copies of S (with one of the holomorphic structures reversed) along


5/44


the boundary; the map exchanging the copies acts on S by an antiholomorphic involution : S S. Similarly, we may glue together two copies of E to a bundle E S. lifts to aninvolution E ofE, so that F is the fixed point set of E. The eigenspace decomposition forE

is

(8) 0(E) = 0(E, F) i 0(E, F)

in any W1,p

norm, p 1, where we identify0(E, F) = { 0(E), E = }.

Given a Cauchy-Riemann operator DE for the double one obtains a Cauchy-Riemann oper-ator DE,F by restriction to

0(E, F). We call the operators obtained in this way odd, sincethe existence of the complex linear extension to S \ S imposes conditions on the 0-th orderpart of the operator at the boundary. The decomposition (8) induces a decomposition ofthe kernel of DE

ker(DE) = ker(DE,F) i ker(DE,F)

and similarly for the adjoints DE

, DE,F. Hence

(9) dim(ker(DE)) = 2 dim(ker(DE,F)), dim(coker(DE)) = 2 dim(coker(DE,F)).

The Chern number of the double E is the Maslov index of F. The Riemann-Roch formulagivesInd(DE) = rankC(E)(S) + 2 deg(E).

The identities(S) = 2(S), I(E, F) = deg(E)

show compatibility of (9) with Riemann-Roch for (E, F) in (5). By Kodaira vanishing, ifTS E is positive then DE is surjective. It follows that DE,F is surjective as well. IfS hasgenus at most one then a line bundle E S is positive if and only if it has positive degree.Hence if S is an annulus, rankC(E) = 1, and I(E, F) > 0 then DE,F is surjective, whileon a disk S it suffices that I(E, F) > 2. Presumably one can generalize these results toarbitrary Cauchy-Riemann operators on surfaces with boundary; we are not aware of anyresults of this type in the literature.

Cauchy-Riemann operators on surfaces with strip-like ends: Let S be a surface with strip-like ends, and E, F a pair of vector bundles as in Definition 4.1.1 of [14]. A real Cauchy-Riemann operator for (E, F) is asymptotically constant if on each strip-like end e E(S)there exists a time-dependent operator

He : [0, 1] EndR(Ee)

such that the operator DE,F on sections

= (e), : R [0, 1] Ee

has asymptotic limit given by

(10) 12d + iEe d j + (He)ds (iEe He)dt.Here iEe and j denote the complex structures on Ee and R

[0, 1] respectively, and dis the trivial connection on the trivial bundle Ee over R

[0, 1]. That is, the differencebetween e

DE,F(e)

and (10) is a zero-th order operator that approaches 0 uniformly in


6/44


all derivatives in t as s . An asymptotically constant Cauchy-Riemann operator DE,Fis non-degenerate if the operator t +He has no kernel. Any non-degenerate, asymptoticallyconstant operator DE,F is Fredholm, see for example Lockhart-McOwen [5] for the case ofsurfaces with cylindrical ends.

Cauchy-Riemann operators on nodal surfaces. A nodal surface S (with boundary and strip-like ends) consists of

(a) A surface with strip-like ends S with boundary S as in Definition 4.1.1 of [14]; inparticular including an ordering of the components of S, the boundary componentsand strip-like ends of each component of S;

(b) An unordered collection of interior nodes: unordered pairs

Z = {z1 , z+1 }, . . . , {z

r , z

+r }

of distinct interior points of S;(c) An ordered collection of boundary nodes: ordered pairs

W = (w1 , w+1 ), . . . , (w

s , w

+s )

of distinct boundary points of S.

Remark 2.3.1. S

is the normalization (resolution of singularities) of S.A complex vector bundle E S on a nodal surface with boundary consists of

(a) A complex vector bundle E S;(b) Isomorphisms E

z+i E

ziand E

w+i E

wifor each interior and boundary node;

(c) A trivialization E|im e= Ee (R

[0, 1]) for each strip-like end e E(S).

A totally real boundary condition F for E S is a totally real subbundle F E|S suchthat:

(a) The identifications of the fibers at the boundary nodes induce isomorphisms Fw+i

Fwi

;

(b) F is maximally totally real, that is rankR(F) = rankC(E

);

(c) In the trivialization over each strip-like end e E(S), the subspaces Fe(s,0) = Fe,0

Ee and Fe(s,1)

= Fe,1 Ee are constant along s R, and they form a transverse

pair Fe,0 Fe,1 = Ee, as in Section 4.1.

Let E S be a complex vector bundle on a nodal surface S with totally real boundarycondition F. By a real Cauchy-Riemann operator DE,F for (S ,E ,F) we mean an operator

DE,F : 0(E, F) 0,1(E, F), DE,F

defined in terms of a real Cauchy-Riemann operator DE,F on S with values in E and

boundary conditions in F. Here we set 0,1(E, F) := 0,1(E, F), define 0(E, F) 0(E, F) as the kernel of the surjective map

: 0(E, F) i

Ez+i

j

Fw+j

i

((z+i ) (zi ))

j

((w+j ) (wj )).


7/44


By a family of nodal surfaces S B we mean a smooth family S B of complexsurfaces (compact, possibly with boundary) over a smooth, open base B, together withnodes Z, W (S)2 varying smoothly over B. By a family of complex vector bundlesE S we mean a complex vector bundle E S, together with smoothly varyingidentifications of the fibers at the nodes and constant trivializations on the strip-like ends.A family of totally real boundary conditions F S consists of a totally real boundary

condition F

S

that is constant in the trivializations on the strip-like ends. A familyof real Cauchy-Riemann operators DE,F for the families (S,E,F) B is a family of realCauchy-Riemann operators Db for (Sb, Eb, Fb), varying smoothly with b B.

The determinant line det(DE,F) for the operator over a nodal surface S is isomorphicto the determinant det(DE,F) for the corresponding operator over the smooth surface S

with resolved nodes by the following construction: Consider the unreduced operator

DunredE,F : 0(E, F)

i

Ez+i

j

Fw+j

0,1(E, F), ((), DE,F).

The kernel and cokernel are canonically isomorphic to those of DE,F, and the isomorphismsdefine an isomorphism of determinant lines

(11) det(DE,F) det(DunredE,F ).

From this we construct the reduced operator

(12) DredE,F : ker(DE,F) i

Ez+i

j

Fw+j

coker(DE,F), ((), 0).

Its kernel and cokernel are canonically isomorphic to those of DunredE,F and the isomorphismsdefine an isomorphism of determinant lines

(13) det(DunredE,F ) det(DredE,F).

Since the domain and codomain of DredE,F are finite dimensional, we have by (3) a canonicalisomorphism

(14) det(Dred

E,F

) maxi

E

z+i

j

F

w+j det(DE,F).

Hence orientations on DE,F and the fibers E

z+i, F

w+jinduce an orientation on DE,F. A

similar isomorphism holds when a surface S and bundles E, F are obtained from anothernodal surface S and bundles E, F by resolving some subset of the nodes of S; that is, byremoving some subset of the sets of interior and boundary nodes Z, W.

2.4. Gluing of Cauchy-Riemann operators. In this section we describe two gluingoperations and construct corresponding isomorphisms of determinant lines. We fix a nodalsurface S with strip-like ends, a complex vector bundle E S, a totally real boundarycondition F S, and a real Cauchy-Riemann operator DE,F as in Section 4.1 of [14].

Gluing of strip-like ends. Let e+

E+

(S) and e

E

(S) be an outgoing resp. incomingend and suppose we are given a complex isomorphism Ee+ Ee mapping Fe+,j to Fe,jfor j = 0, 1. Suppose that the asymptotic limits (10) ofDE,F on the ends e are equal, after

the identification of fibers. Let S = #ee+(S) be the surface formed by gluing the ends of S,


8/44


that is e+(R+ [0, 1]) e(R

[0, 1]) S is replaced by a strip [, ] [0, 1] dependingon a gluing parameter > 0, where {} [0, 1] is identified with e({0} [0, 1]). (This

gluing operation fixes S as a nodal surface with strip-like ends as in Section 4.1 of [14] andDefinition 4.1.1 of [14], up to the choice of a new ordering on the boundary components

and strip-like ends.) Let E, F be the complex vector bundle and totally real boundarycondition over S that arise from gluing E, F via the isomorphism Ee+

= Ee on the middle

strip. Using cutoff functions on the strip-like ends one constructs from DE,F a real Cauchy-Riemann operator DE,F for (S, E, F). For sufficiently large there exist isomorphisms

ker(DE,F) ker(DE,F), coker(DE,F)

coker(DE,F)

defined as follows. Given a section in the kernel of DE,F, one uses cutoff functions on

[, ] to glue it together to a section = # of E S with boundary conditions in F,which is an approximate zero of DE,F. Gluing followed by orthogonal projection onto the

kernel of DE,F defines, for sufficiently large, the isomorphism. The construction for thecokernels follows by identifying the cokernels of DE,F and DE,F with the kernels of theiradjoints. This produces an equality of indices

Ind(DE,F) = Ind(DE,F)

and an isomorphism of determinant lines

(15) det(DE,F) det(DE,F).

Gluing (deformation) of nodes. Consider an interior node ofSrepresented by a pair z S,and R>0+[0, 1]i. Let S be the (possibly still nodal) surface with strip-like ends obtainedby deforming the node, that is, gluing together punctured disks around z using the mapz exp(2)/z. We denote by s + it = ln(z)/ the coordinates on the cylindrical neck

[||, ||] S1

. In the case of a boundary node, we require that the gluing parameter isreal and glue together half-disks by the same map and identify the neck with [ , ] [0, 1]with coordinates s + it. In general, the conformal structure of S depends on the value ofthe gluing parameter , as well as the choices of local coordinates R S1 or R [0, 1] onpunctured neighborhoods of z, which are fixed in the notion of nodes of S. (In addition,one has to choose a new ordering on the nodes and possibly the boundary components of S.)Let E, F denote the vector bundles over S, S obtained by gluing in the trivial bundles(Ez, Fz) = (E

z , F

z) = (E

z+ , F

z+) in the fixed trivialization over the (half-)disks around

z. Using cutoff functions, one constructs from DE,F a real Cauchy-Riemann operator

DE,F for (S, E, F).

The following is a slight modification of [2, Lemma 3.1]; it implies that there is a canon-

ical identification of determinant lines of the deformed Cauchy-Riemann operator with thedeterminant line of the original. We now suppose that the node is on the boundary; theinterior case is similar. We also assume for simplicity that S is smooth.


9/44


Theorem 2.4.1. For sufficiently large values of the gluing parameter there is an exactsequence 1

(16) 0 ker(DE,F)

ker(DE,F)DredE,F Fz coker(DE,F) coker(DE,F) 0

such that

(a) the middle map D

red

E,F has the property D

red

E,F D

red

E,F, the operator of (12), and(b) the map Fz coker(DE,F) is the adjoint of the map coker(DE,F) Fz given by

(0, t)/(ds idt), in the limit , where (0, t) is the midpoint of the neck.

Proof. First we construct suitable Sobolev spaces on the glued surface S, depending on thegluing parameter . Let be a smooth function on S, supported on [6 /7, 6 /7], equalto 1 on [5 /7, 5 /7], with derivative bounded by C/ for some constant C > 0. Considerthe function

(17) C(S), = (1 ) + (e

(s) + e(s+)).

The second term is well-defined since is supported on the neck. Let W1,2 (E, F) theSobolev space with weight function , that is, the space of functions f such that f lies

in W1,2

(

E,

F) the standard Sobolev (1, 2)-space. The Cauchy-Riemann operator DE,Fis Fredholm on this Sobolev space and we have an exact sequence

(18) 0 ker(DE,F) 0(E, F)

0,1(E) coker(DE,F) 0.

Define splittings

(19) 0(E, F) = V ker(DE,F).

(20) 0,1(E, F) = U Fz coker(DE,F)

as follows. Let C(S) denote a slowly varying cut-off function, with (s) = 1 on

the complement of (4 /7, 4 /7) [0, 1] S, equal to 0 on (3 /7, 3 /7) [0, 1] S,and satisfying sup ||, sup |D| < C

1 on the neck (4 /7, 4 /7). For any 0(E, F)

we denote by 0

(

E,

F) the section obtained by multiplying by the cutoff function and using the identification of E and E away from the neck. For sufficiently large,the map

ker(DE,F) ker(DE,F),

is an isomorphism, since the domain is finite-dimensional. Let V denote its W0,2 -orthogonal

complement. Define the first map in (29) to be the composition ker(DE,F) ker(DE,F)

of inclusion and projection along V. We claim that for (1, 0), the restriction ofDE,F to V is uniformly right invertible, that is, there exist constants C and 0 such that

for > 0,

(21) CW1,2

DE,FL2, V.

1In the case that DE,F is odd, the sequence (29) is related to the long exact sequence in algebraic geometry0 ker(DE) ker(DE ) Ez coker(DE) coker(DE) 0 associated to the short exact sequence ofsheaves 0 E E Ez 0, see [4]. If coker(DE,F) = 0, then (29) can be derived from the above longexact sequence, since vanishing of higher cohomology is an open condition in families.


10/44


Suppose otherwise. Then there exists a sequence , V with

(22) W1,2

= 1, lim

DE,FL2= 0.

Denote by S resp. E, F the surface with strip-like ends obtained by removing the nodez, resp. the fibers Ez , Fz . Let S

, E, F be the neck and bundles restricted to the neck.We split into sections supported away from and on the neck, and apply elliptic estimates

for S, S to obtain a contradiction.We claim that the kernel of DE,F may be identified with the kernel of DE,F for any

Sobolev weight (0, 1). Indeed, we may identify S locally with the half-space H. Weassume that our Sobolev spaces on S use a measure that is locally the pull-back of thestandard measure on H. The conformal transformation (s, t) exp(s it) maps theinfinite strip R [0, 1] to H, and the pull-back of the canonical measure on H is e2sdsdt.With our conventions, this is the measure with Sobolev weight = 1. This gives anidentification of the kernel DE,F on W

1,21 (E

, F) with the kernel ofDE,F on W1,2(E, F),

and by elliptic regularity with the kernel of DE,F on W1,2(E, F) for any k. The operator

DE,F Fredholm for weights not in the spectrum Z of the limiting operator on the strip-likeends, see e.g. [5], and the kernel is unchanged by any non-negative perturbation of Sobolev

weight not passing through the spectrum. Hence the kernel ofDE,F on W

1,2

1 (E

, F

) is thekernel on W1,2 (E

, F) for any (1, 0).The kernel of the Cauchy-Riemann operator on the neck DE,F is trivial, by the choice

of Sobolev weights , . The norm of a section of E is comparable to the same sectionconsidered as a section of E, up to a factor e which appears in the second term of (17).Let be a slowly varying function on supported on (2 /7, 2 /7) [0, 1] and equal to1 on (1 /7, 1 /7) [0, 1], and with sup ||, sup |D| < C

1 on (, ). We have forsome constants C > 0 independent of ,

E CE + Ce(1 )E

CDE,FE + C projker(DE,F ) E

+Ce

DE

,F

(1 )E

0

which is a contradiction. The first inequality follows from comparibility of the norms on E,E, and E, the second inequality combines the elliptic estimates for (E, F) and (E, F),and the last uses the bound on the derivative of .

Next we find an approximate description of the image of V under DE,F. Identify

coker(DE,F) with the W1,2 -perpendicular of im(DE,F). Also identify E and E

awayfrom the neck. Define an injection for sufficiently large

coker(DE,F) 0,1(E, F), ;

let coker(DE,F) denote its image. Let Fz the subspace of 0,1(E, F)L2

consisting of

one-forms equal on the neck to fds idt for some f Fz . By multiplying by we obtaina finite-dimensional subspace of 0,1(E, F)L2

, isomorphic to Fz by evaluation at a point

at the mid-point of the neck. For sufficiently large, the sum coker(DE,F) + Fz is


11/44


direct, since the intersection is trivial. Let

U 0,1(E, F)

denote its W1,2 -perpendicular. Let

: 0,1(E, F) U

denote the projection. We claim that DE,F : V U is an isomorphism with uniformlybounded right inverse, for sufficiently large. Otherwise, there is a sequence , V with

(23) V = 1, DE ,FU 0.

The pairing ofDE ,F with any sequence of elements + of T of norm one

approaches zero since cut-off functions are slowly varying. Together with (23) this impliesDE ,F 0 which contradicts (21).

From (18), (19), and (20) for sufficiently large we obtain an exact sequence

0 ker(DE,F) V ker(DE,F) U Fz coker(DE,F) coker(DE,F) 0.

The index equality

Ind(DE,F) = Ind(DE,F) dim(Fz)implies that the restriction ofDE,F to V is an isomorphism onto U. Let Dij , i , j = 1, 2denote the components of DE,F with respect to the splittings above. The kernel ofDE,Fconsists of pairs (1, 2) such that

1 = D111 D122, (D21D

111 D12 + D22)2 = 0.

Hence if we defineDred

E,F= D21D

111 D12 + D22.

then we have an identification

ker(DE,F) ker(DredE,F

), 2 (D111 D122, 2).

The image of DE,F consists of pairs (1, 2) such that 2 D21D111 1 lies in the imageof Dred

E,F. The inclusion ofFz coker(DE,F) into U Fz coker(DE,F) induces an

identification of cokernels of DE,F and DredE,F

, hence the desired exact sequence.

The component of DredE,F

in Fz is given asymptotically by projecting DE,F() onto

Fz. We haveDE,F() (s)(ds + idt).

Pairing with f Fz gives the difference of evaluation maps (z+) (z) paired with f. Itfollows that the limit is

lim

DredE,F

= ((z+) (z), 0) = DredE,F.

To prove the last claim in the theorem, the cokernel of the reduced operator contains asubspace of forms approximately constant on the neck, perpendicular to the image of thedifference of evaluation maps. On the other hand, maps coker(DE,F) to one-formsasymptotically vanishing on the neck.


12/44


The existence of the exact sequence is equivalent to the existence of isomorphisms

(24) ker(DredE,F

) ker(DE,F), coker(DredE,F

) coker(DE,F).

These induce an isomorphism of determinant lines

(25) det(DE,F) det(DredE,F

).

The homotopy of Theorem 2.4.1 induces an isomorphism of determinant lines

(26) det(DredE,F) det(DredE,F

).

Combining (25), (26), (13), and (11) gives our gluing isomorphism

(27) det(DE,F) det(DE,F).

The gluing maps satisfy an associativity property: If S is a nodal surface with strip-likeends and S the surface obtained by deforming two nodes, or deforming one node and gluingtogether two strip-like ends, or gluing together two pairs of strip-like ends, then the resultinggluing isomorphisms det(DE,F) det(DE,F) are independent of the order of gluing. We

consider only the case of two boundary nodes z, z; the cases of interior nodes, strip-likeends, and mixed cases are similar but easier.

Lemma 2.4.2. If denotes the deformation of z and the deformation of z then thediagram

(28)

det(DE,F) det(DE,F)

det(DE ,F ) det(DE, ,F, )?

-

?-

commutes.

Proof. The proof is a minor modification of e.g. [2, Lemma 3.5]. Simultaneous deformationof the two nodes leads to an exact sequence(29)

0 ker(DE, ,F, ) ker(DE, ,F, ) FzFzcoker(DE, ,F, ) coker(DE, ,F, ) 0which induces an isomorphism

(30) det(DE,F) det(DE, ,F, ).

We claim that this isomorphism is equal to the isomorphism given by going either wayaround the square (28). To prove the claim consider the diagram

ker(DE, ,F, ) ker(DE, ,F, ) Fz coker(DE, ,F, ) Fz coker(DE, ,F, )

ker(DE, ,F, ) ker(DE, ,F, ) Fz Fz coker(DE, ,F, ) coker(DE, ,F, )

-Id -

?

-Id

?

-

6

- -

6

For fixed gluing parameters , the diagram commutes up to a small error term whichis irrelevant for the purposes of orientations. The middle maps in the exact sequence


13/44


are, by definition, DE,F and DE,F . By approximate commutativity of the diagram thecomposition of the top and right maps in (28) is equal up to homotopy to (30). A similarargument shows the same for the composition of the two maps on the other side of (28),and this completes the proof.

3. First relative non-Abelian cohomology

IfG is a (possibly non-Abelian) group and M a manifold then the first cohomology groupH1(M, G) for the sheaf of smooth maps to G is well-defined and parameterizes isomorphismclasses of principal G-bundles over M, see for example Serre [11]. This section describesan extension to the simultaneously relative case for a map f : M N and a grouphomomorphism G G/Z given by quotienting by a central subgroup Z. This should alsobe well-known but we were unable to find a reference. In our application, G will be thespin group and Z its center; an element of the relative non-Abelian cohomology group isan isomorphism class of relative spin structures. This definition is equivalent to the oneintroduced in Fukaya-Oh-Ohta-Ono [3], but it makes clear how to parameterize the possiblechoices and avoids the introduction of triangulations. We thank D. Freed for teaching usthe category viewpoint on spin structures.

3.1. Definition via Cech cochains.

Definition 3.1.1. Let Prin(G) denote the category whose objects are principal G-bundlesP M and whose morphisms are G-equivariant isomorphisms P1 P2 covering theidentity on M.

Prin(G) can be understood as follows using non-Abelian Cech cohomology. Let U ={Ui, i I} be a good cover of M, that is, all multiple intersections are contractible. LetCj(M, G) be the set of collections of maps

gi0,...,ij : Ui0 . . . Uij G

and the coboundary operator defined by

: Cj(M, G) Cj+1(M, G), (g)i0,...,ij+1 =

j+1j=0

g(1)j

i0,...,bij,...,ij+1.

The sets Cj(M, G) for j = 0, 1, 2 form a complex in the sense that C0(M, G) acts on theleft on the kernel Z1(M, G) of : C1(M, G) C2(M, G) by the formula

(hg)i0,i1 = hi0gi0,i1h1i1

.

We denote the quotient

H1(M, G) = C0(M, G)\Z1(M, G), H0(M, G) = Z0(M, G).

For G Abelian, the complex extends to j 2 and all cohomology groups Hj(M, G), j =0, 1, 2, . . . are well-defined.

Definition 3.1.2. Let H1(M, G) denote the category whose objects are elements ofZ1(M, G)and whose morphism spaces are C0(M, G).

The set of isomorphism classes of objects is H1(M, G).


14/44


Proposition 3.1.3. Prin(G) is equivalent to the category H1(M, G).

To construct the equivalence, given any principal G-bundle P, choose a local trivializationof P. The transition maps for P form a cocycle c in C1(M, G). Similarly, any morphismfrom P to P defines a chain a in C0(M, G) with a c = c. Conversely define a functor : H1(M, G) Prin(G) by gluing. The bundle obtained by locally trivializing andthen gluing is canonically isomorphic to the original one, and vice-versa, that is, , areequivalences. This also shows that H1(M, G) is independent of the choice of good cover, upto equivalence of categories.

Variation # 1: One can make the construction relative to a smooth map of manifoldsf : M N. We assume that the good cover UM on M is a refinement of the pull-backfUN of the cover UN on N, and we are given a morphism of covers UM fUN, thatis, for each U UM an element V UN such that f(U) V. By pull-back we obtain amorphism of chain groups

f : Cj(N, G) Cj(M, G).

Define

Cj(f, G) := Cj(M, G) Cj+1(N, G), (a, b) = ((a) (fb)(1)j

, b).

For G Abelian the space Cj1(f, G) acts on the space of cocycles Zj(f, G).

Definition 3.1.4. For G an Abelian group, let H1(f, G) be the category whose objects arecocycles Z1(f, G), and whose morphisms are given by Hom(z, z) = {c C0(f, G), cz = z},with composition and identity given by the group structure on C0(f, G).

Remark 3.1.5. An element of Z2(N, G) defines a gerbe, see for example [1]. For G the circlegroup, H1(f, G) is the category ofrelative gerbes discussed in Shahbazi [12] (using a shiftedconvention for degree.)

Variation # 2: One can make the construction relative to group homomorphisms. Anygroup homomorphism : G H induces a functor Prin() : Prin(G) Prin(H), by

the associated bundle construction P (P H)/G where G acts on the right on P andby left multiplication, via , on H. For a principal H-bundle Q, let Prin(G)Q denotethe category of G-structures on Q whose objects are principal G-bundles together with anisomorphism P G H Q, and morphisms are isomorphisms of G-bundles inducing thetrivial automorphism of Q.

We aim to give a cohomological description of Prin(G)Q, in the special case that issurjective and the kernel Z is a central subgroup of G. The short exact sequence of groups

0 Z G H 0

induces a long exact sequence of sets

(31) . . . H 0(M, H) H1(M, Z) H1(M, G) H1(M, H) H2(M, Z).

That is, H1(M, Z) acts transitively on the kernel of H1(M, G) H1(M, H), and the set-theoretic kernel of H1(M, H) H2(M, Z) is equal to the image of H1(M, G). The mapH1(M, H) H2(M, Z) will be called the characteristic class for the short exact sequence.


15/44


Example 3.1.6. Let Spin(r) denote the universal cover of SO(r) and Pin(r) the doublecovers ofO(r) whose centers are Z2 Z2 for Pin(r)

+ and Z4 for Pin(r). The characteristic

classes for Spin(r), Pin+(r), Pin(r) correspond to the Stiefel-Whitney classes w2, w2, w2 +w21 respectively.

Let z Z1(M, H) be a cocycle.

Definition 3.1.7. Let H1

(M,G,z) denote the category of Cech G-structures on z whose objects are cocycles a C1(M, G) with (a) = z; the set of morphisms Hom(a, a) is the set of b C0(M, Z) with ba = a; the identity morphism is the identity 1 C0(M, Z); composition is by group multiplication in C0(M, Z).

Proposition 3.1.8. (a) H1(M,G,z) depends only on the cohomology class of z, up toequivalence of categories;

(b) H1(M,G,z) is non-empty if and only if the image w of z in H2(M, Z) is zero, andif so:

(c) the set of isomorphism classes of objects H1(M,G,z) has a faithful transitive actionof H1(M, Z);

(d) the group of automorphisms of any object is H0

(M, Z).Let Q M be a principal H-bundle represented by a cocycle z Z1(M, H). As before,

choosing local trivializations compatible with those ofQ defines an equivalence of categories

Q : Prin(G)Q H1(M,G,z).

The category H1(M,G,z) can be also be described in terms of trivializations of thecorresponding characteristic class.

Definition 3.1.9. For any cocycle w Z2(M, Z), let H1(M ,Z,w) denote the category oftrivializations of w whose

objects are elements c C1(M, Z) with c = w and the set of morphisms Hom(c, c) is the set of b C0(M, Z) with bc = c;

composition and identity given by the group structure on C0

(M, Z).Proposition 3.1.10. (a) H1(M ,Z,w) is independent underw w(u) up to an equiv-

alence of categories, given by multiplying by u;(b) H1(M ,Z,w) is non-empty if and only if the class of w is zero in H2(M, Z), and if

so:(c) the set of isomorphism classes of objects H1(M ,Z,w) has a faithful transitive action

of H1(M, Z), and(d) the group of automorphisms of any object is H0(M, Z).

Since the set of trivializations of w and the set of lifts of z both have faithful transitiveactions ofC1(M, Z), and the morphisms for both categories are C0(M, Z), we have a (non-canonical) equivalence of categories H1(M,G,z) H1(M ,Z,w).

Example 3.1.11. Taking G = Spin(r), Z = Z2, we obtain that the category of spin structureson an oriented Euclidean vector bundle E is equivalent to the category of trivializations ofits second Stiefel-Whitney class w2(E).


16/44


The first and second variations can b e combined as follows. Let f : M N be a smoothmap of manifolds, Z G a central subgroup of a Lie group G and : G G/Z theprojection.

Definition 3.1.12. Given a cocycle z C1(M, G/Z), let H1(f ,G ,z) denote the categoryof Cech relative G-structures on z whose

objects are cocycles (a, b) Z1(f, G) with (a, b) = (z, 0); note that by definitionb lies in the image of C2(N, Z) in C2(N, G);

the set of morphisms Hom((a, b), (a, b)) is the set of h C0(f, Z) with h(a, b) =(a, b);

the identity and composition are given by the group structure on C0(f, Z).

A relative G-structure on a G/Z-bundle Q M is a relative G-structure (a, b) C1(f, G)on a cocycle z C1(M, G/Z) representing Q. We call the class [b] of b in H2(N, Z) thebackground class of the relative G-structure.

Proposition 3.1.13. (a) H1(f ,G ,z) depends only on the cohomology class ofz inH1(M,G/Z),up to equivalence of categories;

(b) H1(f ,G ,z) is non-empty if and only if the image of the class of z in H2(M, Z) isthe pull-back of a class in H2(N, Z), and if so,

(c) the set of isomorphism classes is H1(f, Z), and(d) the group of automorphisms of any object is H0(f, Z).

Suppose that z C1(M, H) is a cocycle and w C2(M, Z) the coboundary of some liftofz to C1(M, G). Then H1(f ,G ,z) is non-canonically equivalent to the category of relativetrivializations H1(f ,Z,w) of the image w, that is, the category whose objects are cochainsc C1(f, Z) with c = w, and morphisms are chains h C0(f, Z).

Example 3.1.14. For G = Spin(r), Z = Z2, the category of relative spin structures on aEuclidean vector bundle E is non-canonically equivalent to the category of relative trivial-izations of its second Stiefel-Whitney class w2(E).

The following is included to connect the Cech definition with that of [3].

Proposition 3.1.15. Suppose thatQ M is a G/Z-bundle andR N aG/Z-bundle withcharacteristic class pulling back under f to that ofQ. There is a one-to-one correspondencebetween relative G-structures on Q and G Z G-structures on Q fR.

Proof. Suppose that (a, b) C1(M, G) C2(N, Z) is a relative G-structure on Q. Let c C1(N,G/Z) be a cocycle representing R, mapping to b C2(N, Z) under the coboundarymap. By definition c has a lift d C1(N, G) such that d = b + e, for some e C1(M, Z).The chain (a, fde) C1(M, G)C1(M, G) = C1(M, GG) has boundary (fb, fb) C2(M, ZZ). Quotienting by the antidiagonal action of Z defines a cocycle in C1(M, GZG) whose image in C1(M,G/Z G/Z) represents Q fR. Conversely, given a G Z G-structure on QfR any lift of the form (a, fd) C1(M, GG) must satisfy a = fd,and so defines a relative G-structure on Q with b = fd.

In particular, if E M has a relative spin structure and V N is a bundle withw2(V) restricting to w2(E), then a relative spin structure is equivalent to a spin structureon E fV, as in [3].


17/44


3.2. Definition via classifying spaces. In this section, we describe an alternative ap-proach using classifying maps. Let G be a compact group, and EG BG the universalbundle. Let [M,BG] denote the space of homotopy classes of maps M BG. Assigning toany principal G-bundle P the homotopy class of a classifying map for P defines a bijectionH1(M, G) [M,BG].

Variation # 1: Homotopy relative G-bundles for a smooth map . Suppose that G is Abelianand f : M N is a smooth map. The G-bundle EG G EG BG BG induces a mapBG BG BG giving BG the structure of an H-space. Let B2G denote its classifyingspace. A homotopy relative G-bundle is a homotopy class of a pair (, ) consisting of amap : N B2G together with a section of the pullback BG-bundle : fE(BG).

Variation # 2: Homotopy relative G-bundles for a group homomorphism. Suppose that 0 Z G G/Z 0 is a short exact sequence of groups, with Z a central subgroup ofG. Wehave an induced fibration BZ BG B(G/Z) of classifying spaces. Given a G/Z bundleQ, we denote by [M,BG]Q the space of homotopy classes of maps whose composition to amap to B(G/Z) corresponds to Q. An element of [M,BG]Q is called a homotopyG-structureon Q. The classifying map construction induces a bijection H1(M, G)Q [M,BG]Q.

One can combine the first two variations as follows. A homotopy relative G-structure on aG/Z-torsor Q M with a classifying map M B(G/Z) is a homotopy class of a pair (, )consisting of a map : N B2Z together with a section of the associated BG-bundle(fE(BZ)) BZBG, such that the associated section of the trivial B(G/Z)-bundle is thegiven classifying map for Q.

The following relates the construction to trivializations. A homotopy relative trivializationof a map : M B2Z is a homotopy class of a pair (, ), consisting of a map : N B2Zand a section of the BZ-bundle (f) : M B2Z. Since Z is Abelian, the fibrationBZ BG B(G/Z) is a BZ-torsor and induces a map B(G/Z) B2Z, where B2Z =B(BZ) is an Eilenberg-Maclane space classifying second cohomology with coefficients in Z.

Proposition 3.2.1. IfG is simply-connected, then the set of homotopy relative G-structureson a bundle given by a map M B(G/Z) is in bijection with the set of homotopy relativetrivializations of the associated characteristic class M B2Z.

Proof. Recall the definition of Postnikov truncation: Given a space X with the homotopytype of a CW-complex, the Postnikov tower for X is a sequence

. . . X n Xn1 . . . X1

constructed from X inductively by attaching cells to kill the higher dimensional homotopygroups, so that

j(Xn) =

j(Xn) j n0 otherwise

.

The Postnikov construction is functorial and so induces maps

BZ (BG)3 (B(G/Z))3 B2Z.

Suppose that G is 1-connected. Then BG is 2-connected, hence (BG)3 is trivial and soB(G/Z)3 and B

2Z are homotopic. In this case, any G-structure on Q defines a trivializationof the characteristic class .


18/44


More generally, suppose that f : M N is a smooth map and a G/Z-bundle Q isequipped with a homotopy relative G-structure. The BZ-torsor : N B2Z induces amap of sections

[N,BZ] [N, E(BZ)] [N, B(G/Z)] [N, B2Z]

where the last map is induced by the map B(G/Z) B2Z, followed by multiplication by

. We can now apply Postnikov truncation to the fibers, to get a sequence[N,BZ] [N, 3] [N, (B(G/Z))3] [N, B

2Z].

Pulling back to M we get a sequence

[M,BZ] [M, f3] [M, B(G/Z))3] [M, B2Z].

where the last map is the composition of B(G/Z)3 B2Z with multiplication by f.

Now since BG3 is a homotopy point, f3 is also and any relative G-structure gives rise to

a trivialization of f, that is, a relative trivialization of . Since the higher homotopygroups of B2Z vanish, this correspondence is a bijection.

The extension of these notions to group bundles is left to the reader.

3.3. Relative spin structures.

3.3.1. Operations on relative spin structures. Relative spin structures were defined in Exam-ple 3.1.14. Let V, W be oriented Euclidean vector spaces and SO(V), SO(W) their groupsof automorphisms. The group homomorphisms SO(V)SO(W) SO(V W), SO(V) SO(V) lift to the corresponding Spin groups and induce functors

Prin(Spin(V)) Prin(Spin(W)) Prin(Spin(V W))

Prin(Spin(V)) Prin(Spin(V)).

These operations extend to relative spin structures as functors

H1(f, Spin(V))E H1(f, Spin(W))F H1(f, Spin(V W))EF

H1(f, Spin(V))E H1(f, Spin(V))E.

We will also need the following doubling construction. If E M is an oriented vectorbundle then the direct sum EEhas a canonical spin structure, induced from the canonicallift of the diagonal embedding SO(V) SO(V V) to Spin(V V). That is, there is acanonical functor

(32) Prin(SO(V)) Prin(Spin(V V)).

3.3.2. Classification theorem on surfaces.

Proposition 3.3.1. Suppose that S is a compact, oriented surface with boundary S, andQ S is an SO(r)-bundle. There is a bijection between isomorphism classes of relativespin structures on Q for the inclusion S S, homotopy classes of stable trivializations ofQ, and isomorphism classes of stable spin structures on Q.


19/44


Proof. Let f : S S be the inclusion of the b oundary. Since S is two-dimensional,any cohomology class w H2(S,Z2) is the second Stiefel-Whitney class of some bundle(since the third Postnikov truncation of B Spin is BZ2). From Proposition 3.1.15 (or thehomotopy definition) we obtain a bundle R S together with a spin structure on Q fR.We may assume that S is non-empty, since otherwise the statement is vacuous. Thus Sis homotopy equivalent to a b ouquet of circles. Since 2(S) is trivial, the bundle R S

is trivial and so the relative spin structure gives a stable trivialization of S. If S is a disk,then the trivialization of R (and therefore also the stable trivialization of S) is uniqueup to homotopy. In general, two stable trivializations differ by a map S SO. Since[S,SO] = [S, (SO)2] = H

1(S,Z2), there is no longer a distinguished stable trivialization.However, the image of H1(S,Z2) H

1(S, Z2) is trivial, which implies that fR has a

distinguished trivialization. Hence S has a distinguished stable trivialization. Conversely,any stable trivialization of S induces a relative spin structure (by taking R to be thetrivial) bundle and this gives the first bijection. The second bijection is well-known; itfollows since Spin() is 2-connected implies that B Spin() is 3-connected, and so anymap S B Spin() is homotopic to a constant map, by a homotopy that is unique upto homotopy of homotopies.

4. Orientations for Cauchy-Riemann operators, continued

In this section we define orientations for Cauchy-Riemann operators from an orientationand relative spin structure on the totally real boundary condition, and investigate theirbehavior under gluing.

4.1. Construction of orientations.

Proposition 4.1.1. Suppose that S B is a family of nodal surfaces without strip-likeends, (E, F) B is a family of complex vector bundles E S with oriented totally realboundary conditions F E|S, and DE,F a family of real Cauchy-Riemann operators for(S ,E ,F). A relative spin structure for the bundle F S, if it exists, defines an orientationfor the determinant line bundle det(DE,F) B.

Here B is a smooth open base, so S =

bB Sb is a bundle over B whose fibres are theboundaries of the fibres of S.

Proof of Proposition 4.1.1.Orientations for families of smooth, closed surfaces: Suppose that S B is a family ofsmooth surfaces without b oundary or strip-like ends. Consider a family DE of real Cauchy-Riemann operators acting on sections of a family of complex vector bundles E S. Sincethe space of real Cauchy-Riemann operators is an affine space containing the complex linearCauchy-Riemann operators, there exists a homotopy from DE to a family of complex linearoperators DE. The complex structure on the kernels and cokernels of D

E induce orientations

for DE, which pull back under the isomorphism of determinant lines det(DE) det(DE) to

orientations ofDE. Any two homotopies are related by a homotopy of homotopies, since thespaces of real Cauchy-Riemann operators and complex-linear Cauchy-Riemann operatorsare contractible. Hence the orientation on det(DE) is independent of the choice of D

E and

homotopy to DE.


20/44


Figure 1. Pinching off a set of disks

Orientations for smooth, compact surfaces with boundary: Suppose that B = {pt} and S is

a smooth, compact surface with boundary. From the relative spin structure on F we obtaina stable trivialization of F S, by Proposition 3.3.1. Since 0(O(n)) = Z2 for any n, Fadmits a trivialization.

First we fix a trivialization of F and construct an orientation for DE,F; later we willshow that the orientation depends only on the homotopy class of stable trivializations.The real Cauchy-Riemann operator DE,F acts on sections of E S with totally realboundary conditions F E|S. The trivialization F = R

n S induces a trivializationE|S = F iF = C

n S, which extends to a neighborhood U S of S. Deform

the complex structure on S to a nodal surface S by pinching off a disk for each boundarycomponent, as follows. Choose the neighbourhood U = iUi S as disjoint union of annuliUi = [1, 1] S

1 with S Ui = {1} S1. Replacing Ui with annuli of increasing radius

produces a family of surfaces, whose limit is the nodal surface obtained by replacing Ui with

two disks Di D

+i glued at an interior node {z

i , z

+i }, z

i = 0 D

i , z

+i = 0 D

+i . Here

D+i is the unit disk with standard complex structure jstd and b oundary D+i identified with

{1} S1 Ui, whereas Di is the unit disk with complex structure jstd and boundary

Di identified with {1} S1 Ui. So the nodal surface S is given by the resolution

S = Smain Sdisk, consisting of a closed surface Smain = (S \ U) iDi and a union of

disks Sdisk = iD+i , and a collection of interior nodes Z = {{z

i , z

+i }} between z

i Smain

and z+i Sdisk, see Figure 1. This pinching also induces a complex vector bundle E S

and totally real boundary condition F as follows: Let Emain Smain be the complexvector bundle defined by gluing together E|S\U (which is trivialized = C

n iDi on the

boundary) with the trivial bundle on iDi . Let Edisk denote the trivial bundle C

n iD+i

and F Edisk|Sdisk the trivial bundle Rn iD

+i

= F. Then E S is given by

E := Emain Edisk S and identification at the nodes Z, and F = Fdisk is given by F.Conversely, (S ,E ,F) is obtained from (S, E, F) by gluing at the interior nodes. So by (27)we have an isomorphism of determinant lines det(DE,F) det(DE,F), and combined with


21/44


(11), (13), and (14) we obtain an isomorphism

(33) det(DE,F) max

i

Ez+i

det(DE,F).

Here the first factor is oriented by the complex structure on Ez+i

, and the second factor

decomposes into det(DE,F) = det(DEmain DEdisk,Fdisk) = det(DEmain) det(DEdisk,Fdisk).The operator DEmain has an orientation given by the previous step, since Smain is smoothand closed. On the other hand, by construction the operator DEdisk,Fdisk is the direct sum

of real Cauchy-Riemann operators on the standard bundles (Cn,Rn) over the disk. Aftera homotopy, these are the standard Cauchy-Riemann operators, which are surjective andwhose kernel ker DEdisk,Fdisk = iR

n is isomorphic to a sum of fibers iFzi of Fdisk= F,

via evaluation at points si D+i S on the boundary. The orientation of the boundary

condition F (induced by the trivialization) thus defines an orientation on DEdisk,Fdisk. The

orientation on DE,F is induced from the isomorphism (33).The construction of the orientation involved several auxiliary choices: the trivialization

ofF, the extension of the induced trivialization of E to the neighborhood U, and the choice

of coordinates on U. Any two choices of extensions and coordinates on U are homotopic.Any two trivializations of F S differ by a map : S SO(rank(F)). Hence there aretwo trivializations up to homotopy for each boundary component if rank(F) > 2, infinitelymany if rank(F) = 2, and a unique trivialization if rank(F) = 1. This means that there aretwo stable homotopy classes of stable trivializations of F, for any rank. We claim that theorientation on DE,F depends only on the stable homotopy equivalence class of the stabletrivialization defined by the chosen trivialization of F.

First, consider two choices of extensions and coordinates, and a homotopic pair of triv-ializations of F. From the homotopies we obtain continuous families of nodal surfacesand bundles St, Et, Ft, Cauchy-Riemann operators DEt,Ft, and isomorphisms det(DE,F)

det(DEt,Ft) for t [0, 1]. The construction fixes an orientation for each det(DEt,Ft) from

the orientations for the nodal fibres (

E

t )zi+(t), the operators D(Emain)t on complex bundlesover closed surfaces, and the operators D(Edisk)t,(Fdisk)t on trivial bundles over disks. Each

of these orientations is continuous in families, hence the orientations on det( DEt,Ft) vary

continuously in t. It follows that the map det(DE0,F0) det(DE1,F1) induced by the homo-

topy of operators (DEt,Ft)t[0,1] preserves the given orientations. The composition of this

map with det(DE,F) det(DE0,F0) is homotopic to det(DE,F) det(DE1,F1), and hence

the two isomorphisms induce the same orientation on det(DE,F).Finally, it remains to check that trivializations of F which are homotopic after stabiliza-

tion also define the same orientation on DE,F. (For rank(F) > 2 there is nothing to show,since the trivializations are homotopic iff they are stably homotopic.) Let Ftriv be the trivialRk-bundle over S, Etriv the trivial C

k-bundle over S, and consider two trivializations of F

such that the induced trivializations ofFtriv F are homotopic. By the previous discussionthese trivializations define the same orientation for DEtrivE,FtrivF := DEtriv,Ftriv DE,F,where DEtriv,Ftriv is the standard Cauchy-Riemann operator. On the other hand, ourconstruction using the canonical trivialization for Ftriv and a given trivialization for F


22/44


separately, and then applying the direct sum isomorphism (1) provides an orientation ofdet(DEtriv,Ftriv) det(DE,F)

= det(DEtrivE,FtrivF), which is related to the previous oneby a universal sign, that only depends on the combinatorics of the surface and the ranksof the bundles. Since the first orientation was the same for both trivializations of F, theorientation for DE,F must also be the same for both trivializations.

Orientations for families of smooth, compact surfaces with boundary: It suffices to show thatthe orientations for det(DE,F)b, b B constructed above vary continuously in B. For this itsuffices to consider family S B of smooth surfaces with B contractible. A trivializationof F S induces a trivialization of E near the boundary S. Deforming the conformalstructure on S B as in the previous step produces a family of nodal surfaces S B,which consists of a disk bundle Sdisk B, a family of closed surfaces Smain B (obtained

by gluing a disk bundle to S), and identifications of Sdisk and Smain at families of interiornodes. This deformation provides an isomorphism of determinant line bundles

det(DE,F) det(DE,F) max

i

Ez+i

det(DEmain) det(DEdisk,Fdisk),

which defines the orientation on det(DE,F) by pullback from the right hand side. These

orientations vary continuously: The orientation on the first factor is induced from thecomplex structure on E

z+i B, on the second factor it is given by the previous construction

for families of closed surfaces, and the third factor is isomorphic (using a homotopy to thestandard Cauchy-Riemann operator on disks) to det(iFzi) for a smooth family of boundarypoints si S in each connected component. These fibres of F S are oriented byassumption, inducing a continuous orientation on det(DEdisk,Fdisk) and hence on det(DE,F).

General definition of orientations: Finally, we consider a general family of nodal (butcompact) surfaces S B and real Cauchy-Riemann operators DE,F for families of complexvector bundles E S and totally real boundary conditions F. This family of operators isgiven by identifications of families of nodes from a family of real Cauchy-Riemann operatorsDE,F for families of bundles E

S and F S over the family of smooth resolutions

S B. We fix a trivialization ofF, that is a trivialization ofF S which is compatiblewith the identifications at nodes. From (11), (13), and (14) we have a bundle isomorphism

(34) det(DE,F) max

i

E

z+ij

F

w+j

det(DE,F)

Here an orientation on DE,F is given by the previous step, the complex fibers of E arenaturally oriented, and the fibers ofF are oriented by assumption. Hence this isomorphismdefines orientations on DE,F.

Now we construct orientations for Cauchy-Riemann operators on surfaces with strip-like ends. Fix S1 to be the once-punctured disk with a complex structure such that aneighbourhood of the puncture corresponds to an incoming strip-like end. We identify its

boundary S1 = R, preserving the orientation.

Definition 4.1.2. Let (E, F, F+, H) be a tuple consisting of a complex vector space E anda pair (F, F+) of transverse, oriented, totally real subspaces of half-dimension, equipped


23/44


with spin structures, and H a normal form for a Cauchy-Riemann operator on the strip asin (10). An orientation for (E, F, F+, H) consists of

(a) a smooth path : R Real(E) of totally real subspaces connecting () = F;it is identified with a totally real boundary condition E S1 for the trivialbundle E S1,

(b) a real Cauchy-Riemann operator D on S1 for sections with values in the trivial

bundle E and boundary values in , with asymptotic limit given by H;(c) an orientation for D;(d) a spin structure on , extending the given spin structures on F, F+.

Let S be a surface with strip-like ends, S the surface obtained by adding the points atinfinity, E S a complex vector bundle and F E|S totally real boundary conditions.For each end e E(S), the corresponding point at infinity is ze S, and the two realboundary conditions meeting it are Fe, Fe1. By assumption, these are constant transversesubspaces, Fe,0Fe,1 = Ee over the strip-like end. We suppose that that we have chosen spinstructures on the fibers Fe,0, Fe,1 at infinity, and set of asymptotic limits H = (He, e E(S)).

Definition 4.1.3. An orientationfor (S,E,F, H) is a tuple of disk orientations (e, De, oe, Spin(e))for (Ee, Fe,0, Fe,1, He) for each e E(S).

By a relative spin structure on F, we mean a stable spin structure extending the givenstable spin structures on Fe,0, Fe,1 at infinity on each end e E(S).

Proposition 4.1.4. Let S B be a family of nodal surfaces with strip like ends, andE B a family of complex vector bundles with totally real boundary conditions F B.A choice of a relative spin structure on F, if it exists, and orientations for the ends of(S,E,F, H) induce an orientation of the determinant line bundle det(DE,F) B.

Proof. First consider the case that the boundary ofS is connected. On each strip-like end econsider the deformation of the boundary conditions Fe, on a neighborhood of infinity tothe boundary condition formed by concatenating the restriction of with 1, which has acanonical deformation to the boundary condition with constant value (). The resulting

boundary value problem is obtained by deformation of the nodes of a nodal surface S withvector bundles E, F obtained by gluing together the problems (Ee, Fe,0, Fe,1, He) and aproblem (D,E ,F) on a closed (possibly nodal) surface obtained by gluing (Ee, Fe,0, Fe,1)

onto (E, F), see Figure 2. The nodal surface S has a canonical order of components givenby taking the ordering of the additional components to be the one given by the strip-likeends, and ordering of the boundary nodes so that the original component is ordered first.Let (E, F) denote the vector bundles on the nodal surface. (15) gives an isomorphism ofdeterminant lines

(35) det(D) det(D).

From (14) we obtain an identification

(36) det(D) eE

det(De ) max(e(0)

) det(D) eE+

max(e(0))

eE+

det(D+e )

where e(0) is the fiber given by evaluation the corresponding path e at 0, and the orderof the two products over E is reversed. This choice of order means that when gluing, we


24/44


Figure 2. Pinching off the strip-like ends

Figure 3. Ordering of determinant lines

obtain a product over ends of an incoming end, and outgoing end, and a determinant line ofa dualized fiber, each of which is canonically trivial leaving a product of determinant linesof dualized fibers. Note that we have chosen an ordering for the determinant lines whichis not a special case of the convention for nodal surfaces. The relative spin structure on Fand the bundles e define a spin structure on F, hence an orientation on the correspondingindex by Proposition 4.1.1.

In the case that S is disconnected, we define the orientation on S as a product of ex-pressions in the right hand side of (36), ordering the nodes at the outgoing ends after thedeterminant line of the closed surface corresponding to each boundary component, andeach group consisting of determinant line for a closed surface and its outgoing nodes, in thereverse of the given order on boundary components. See Figure 3 for the ordering of thedeterminant lines for a surface with strip-like ends with two components.

ing fig

4.2. Effect of re-ordering on orientations. The following theorem describes the effectof changing the ordering of boundary components, boundary nodes, or strip-like ends.

Theorem 4.2.1. (Behavior of orientations under reordering) Let S be a nodal surface withstrip-like ends and DE,F a Cauchy-Riemann operator.

(a) Suppose thatS is a nodal surface obtained by re-ordering a boundary node (w+, w)

(w, w+), and DE,F is a Cauchy-Riemann operator obtained from DE,F. The iso-morphism det(DE,F) det(D

E,F) of determinant lines induced by the isomorphism

of kernel and cokernel acts on orientations by (1)rank(F).(b) Suppose that S is a nodal surface obtained by transposing two components Si, Sj of

S. The isomorphism det(DE,F) det(DE,F) of determinant lines induced by the

isomorphism of kernel and cokernel acts on orientations by(1)Ind(DEi,Fi) Ind(DEj,Fj ).

(c) Suppose that S is a nodal surface obtained by re-ordering the boundary compo-nents (resp. boundary nodes) by a permutation . The isomorphism det(DE,F) det(DE,F) of determinant lines induced by the isomorphism of kernel and cokernel

acts on orientations by det()rank(F).(d) Suppose that S is a nodal surface obtained by transposing a pair e, e of consec-

utive strip-like ends. The isomorphism det(DE,F) det(DE,F) of determinantlines induced by the isomorphism of kernel and cokernel acts on orientations by(1)Ind(De) Ind(De ).


25/44


Proof. By the behavior of determinant lines under permutations (2), the behavior of theisomorphism with the trivial determinant (3), and the definition of the orientation on nodalsurfaces (14).

4.3. Effect of gluing on orientations. We now determine the signs induced by the gluingof Cauchy-Riemann operators. Let S be a nodal surface with strip-like ends, and S a nodal

surface obtained by either deforming away a boundary node, deforming away an interiornode, or gluing together two strip-like ends. Let E be a complex vector bundle with totallyreal boundary condition F, and E, F the vector bundles on F obtained by gluing. Similarlylet DE,F be a real Cauchy-Riemann operator with non-degenerate limits along the strip-like

ends, and DE,F an operator obtained by gluing. In the case that S is obtained by gluing

together two strip-like ends e, we assume that the limits He along the glued ends eare equal after identification of the fibers. Let De denote Cauchy-Riemann operators onthe caps Se added to the outgoing and incoming ends in (35). By gluing together the

caps Se we obtain a surface S homeomorphic to the disk with pair (E, F) of Maslov index

zero. By the previous construction the Cauchy-Riemann operator De is equipped with anorientation.

Definition 4.3.1. We say that the orientations are chosen compatibly if the orientationson De have been chosen so that the gluing isomorphism

(37) det(De) det(De+) det(De)

is orientation preserving.

Suppose that a surface S is obtained from S by gluing. In the case of gluing boundarynodes or strip-like ends we assume that the boundary components joined by the gluing areadjacent in ordering; then we give the boundary components of S the ordering obtained byinserting the new boundary component(s) in place of the old in the ordered sequence. Thefollowing theorem describes the effect of gluing on orientations, in this special case.

Theorem 4.3.2. (Behavior of orientations under gluing)

(a) For smooth deformation (gluing) at interior nodes, the isomorphism of determinantlines constructed in Section 2.4 is orientation preserving,

(b) For smooth deformation (gluing) at a boundary node for a nodal surface with a singlenode (w+, w) joining two distinct boundary components adjacent in ordering (resp.a node (w+, w) joining a single boundary component) the isomorphism of deter-minant lines constructed in Section 2.4 acts on orientations by a sign (1)rank(F),with positive sign iff the ordering of w, w+ agrees with the ordering of the boundarycomponents for the pre-glued surface (reps. the ordering of the boundary componentsof the glued surface has the boundary component corresponding to the segment fromw to w+ ordered first.)

(c) For gluing of strip-like ends of distinct components S, S+, assume that S have

connected boundary, the orientations on De have been chosen compatibly, the ende+ is the last outgoing end of S and the end e is the first incoming end of S+,and the ordering of the ends on the glued surface is induced by the ordering of endson S, S+. Then isomorphism of determinant lines constructed in Section 2.4 acts


26/44


on orientations by a sign (1)rank(F) with positive sign in (1) iff the ordering ofe, e+ is (e, e+), times

(1)(P

eE(S+){e}rank(F)Ind(De))(

PfE+(S){e+}

rank(F)Ind(Df))

times

(1)(P

fE+(S+)rank(F))(

PfE+(S){e+}

rank(F)Ind(Df)).

In particular, for one outgoing end or one incoming end and ordering (e, e+), thegluing sign is positive.

(d) If the outgoing ends of S are simultaneously glued to the incoming ends of S, andthe orientations are chosen compatibly then the gluing sign is equal to the sign forgluing of S, S+ along boundary nodes, one for each strip-like end.

Proof. Interior Gluing: Let S denote the smooth surface associated to S, and E, F thecorresponding vector bundles. First we assume that S has empty boundary. A deformationof DE,F to a complex-linear operator produces a similar deformation for the smooth

surface and bundles associated to S. The gluing isomorphism (27) induces an identificationof determinant lines for each bundle in the homotopy. Since the identification of determinantlines for the complex-linear operators is complex linear, the identification of determinant

lines is orientation-preserving, for each bundle in the homotopy.IfS has non-empty boundary, a deformation of (E, F) to the connect sum of a problem

on a closed surface Smain, glued to a trivial problem on a union of disks Sdisk, induces acorresponding deformation for the glued problem (E, F). Compatibility of orientations forgluing closed surfaces implies that the gluing map is orientation preserving.

Boundary Gluing: Suppose that (w+, w) is a boundary node ofS, and S, E, F a surface andbundles obtained by deforming the node. Consider the diagram of indices shown in Figure4; for self-gluing of a disk, see also Figure 5. 1, 2 are the gluing maps for the determinantlines for Smain Sdisk to those of S and Smain Sdisk to S, and are orientation preservingby definition. The surface Sdeg at bottom left is obtained by first gluing at the boundary,and then degenerating the circles used for the degeneration of S. The gluing map 3 for

Sdeg,+ Sdeg, to Sdeg is orientation preserving by definition. The map represents gluingof a collection of disks equipped with trivialized boundary condition, while , 2 representgluing at an interior node and so are orientation preserving by the previous section. Boththe lower square and the upper left triangle in the diagram commute by associativity ofgluing in Section 2.4. Therefore, the map 1 representing gluing of determinant lines fromS to S, induces the same sign on orientations as .

Suppose that S is obtained from a pair S of disks by joining them at a boundary nodew. The operator DE,F has kernel isomorphic to Fw Fw (via the two evaluation mapson the boundary) and vanishing cokernel. The reduced operator is

(38) DredE,F : Fw Fw Fw, (f, f) f+ f.

The ordering of the factors is determined by the ordering of the boundary components of S.By (3) and (25) the induced map det(DredE,F) det(DE,F) changes the defined orientations

by a sign (S, S, rank(F)) = (1)rank(F) depending on whether the ordering of the pair w

agrees with the ordering of the boundary components of S.


27/44


1

2

2

1

3

Figure 4. Gluing at a boundary point

Suppose that S is a single disk, joined to itself via a pair of points w on the boundary.The boundary component is split into two, as in Figure 5. On the normalization S we have

Figure 5. Gluing a disk to itself

ker(DE,F) isomorphic to Fw via evaluation at a boundary point and trivial cokernel. Thereduced operator is

(39) DredE,F : Fw Fw , f 0.

The kernel is isomorphic to Fw and the cokernel is isomorphic to F

w. The deformed

surface S is an annulus, equipped with trivial bundles E, F. The orientation for DE,F isinduced from pinching off a pair of disks, so that S2 is obtained by joining two disks anda sphere at interior points. One has to choose the ordering of boundary components on S;


28/44


this induces an ordering of the nodes z, z+ of S2. On the normalization S2 the reduced

operator can be identified with

DredE2,F2 : Fz Ez Fz+ Ez Ez+, (f, e , f +) (f e, f+ e).

The kernel is isomorphic to Fz , via evaluation at any boundary point, and the cokernel isisomorphic to Ez/Fz = iFz , via projection onto the second factor of the codomain. Let

e1, . . . , en, f1, . . . , f 2n, g1, . . . , gn Fz+ Ez Fz

be a basis for the domain and

h1, . . . , h2n, i1, . . . , i2n Ez Ez+

a basis for the codomain so that

DredE2,F2 : ej hj, fj hj ij , gj ij.

The isomorphism (3) induces for the zero operator the orientation defined by the basis

(40) i2n . . . in+1 (h2n)

. . . (hn+1)

(in) . . . (i1)

hn . . . h1 e1 . . . en f1 . . . f2n

(e1 + f1 + g1) . . . (en + fn + gn).

This differs from the standard orientation for the zero operator by a sign (1)rank(F)2

=(1)rank(F). In order to compare the two orientations coming from deforming the nodalsurfaces S, S2, we compare the identifications of the kernel and cokernel with Fw, Fw(resp. Fw , iFw .) The isomorphisms of the kernel with Fw are both given by evaluationat a boundary point, and so identical. The isomorphism of the cokernel with Fw is givenby evaluation at a boundary point on the strip-like neck, see Figure 6, for S, and for a pointon one of the cylindrical necks, for S2. By construction, the bundle E is trivial. Choose a

Figure 6. Two kinds of neck

homotopy between the two conformal structures on the annulus. Taking the trivial bundleover the homotopy, we obtain a family of indices each with kernel and cokernel isomorphic toFw. We can also deform the evaluation maps to all lie on the boundary, without changing

the induced orientations. It remains to compare the various trivializations of the cotangentbundle used in (29). For the surface S, the local coordinates depend on the ordering ofthe pair w. In the Figure we suppose that w resp. w+ is the point on the left resp.right of the neck. Thus the local coordinate is (in the coordinates s, t on the page) s + it


29/44


if w is numbered first, and s it if w+ is numbered second. On the other hand, thecoordinates on the cylindrical neck are (on the intersection of the two necks) t is. Thusthe second trivializations is related to the first by composition with multiplication by iif w is ordered first. It follows that the gluing map acts by the sign (1)

rank(F), if w isordered first.

Gluing of strip-like ends: First, we consider the case of a disconnected surface S = S S+with a single pair of strip-like ends, and E S a complex vector bundle over S equippedwith totally real boundary conditions F. Given ends e, e+ and an identification of thecorresponding fibers (Ee+, Fe+) (Ee, Fe), let S denote the surface obtained by gluing

S together along the ends, and (E, F) the elliptic boundary value problem obtained bygluing E, F. See for example Figure 3 of [14]. Adding in the points at infinity gives surfaceswithout strip-like ends

S = Se

se, S = S

e=e

se.

Choose an ordering of the boundary components of S. The strip-like ends of S inherit anordering from the ordering of the ends of S. We claim that the isomorphism of determinantlines from S to S has the same sign as the isomorphisms of determinant lines from S to

S. Consider the diagram of indices shown in Figure 7. The top left picture representsdet(DE,F). The map 1 represents the isomorphism of determinant lines induced by de-forming the boundary conditions F as above. The map 2 represents the isomorphismof determinant lines induced by gluing on the orientations. The maps 3, 4, 5 are gluingisomorphisms for the gluing of strip-like ends. The map 6 again represents a deformation,and 7 the isomorphism induced by gluing along two points on the boundary. The firstsquare in the diagram commutes because deformation commutes with gluing; the secondby associativity of gluing for determinant lines. By definition the composition of 1, 2 is

1 2

3 4 5

76

Figure 7. Orientations for gluing strip-like ends

orientation preserving. 5 is orientation preserving by construction, and 6 is orientationpreserving since it is induced by a deformation. Hence 3 has the same sign as 7. By

definition 7 is the composition of gluing isomorphisms for resolution of the first, then sec-ond boundary node. By the discussion above, by choosing the ordering of the boundarycomponents so that the disk boundary is ordered first and boundary nodes so that the nodeon the disk is ordered first, we can insure that the first gluing isomorphism is orientation


30/44


preserving, and the resulting surface is S. Hence 3 has the same sign as the isomorphismof determinants induced by the second gluing operation. By part (b), this has the signclaimed in the statement of part (c).

The additional signs in the case of multiple ends arise from permuting the remainingoutgoing ends and nodes of the S, with the incoming ends and nodes of S+, and alsothe outgoing ends of S with the nodes associated to the outgoing ends of S+, per the

convention (36). Note that the sign resulting from permuting the remaining incoming endsof S+ past the caps and nodes for e and determinant line on the closed surface S is 1,

being (1)3 rank(F)+Ind(De )+Ind(De+) = (1)4 rank(F).

4.4. Orientations for quilted Cauchy-Riemann operators. Recall from [14] that aquilted surface with strip-like ends is a collection of surfaces S with strip-like ends withsome boundary components identified. Let S B be a family of quilted surfaces possiblywith strip-like ends and (E, F) S a family of complex vector bundles over the componentstogether with totally real subbundles over the boundary components and seams. Let DE,Fbe a family of Cauchy-Riemann operators for (E, F). The basic idea of the constructionof orientation for DE,F is to deform everything to split form. There are two steps: firstthe deformation of the operators and boundary and seam conditions at infinity along thestrip-like ends, and then the boundary and seam conditions in the interior. Since theconditions at infinity are constant over the base B, it suffices to show the existence of somedeformation; of course the orientations constructed will depend on this choice. One firstdeforms the asymptotic operators to ones with trivial zero-th order term, through a familyof non-degenerate operators. (That is, one turns the perturbation off, while keeping theboundary and seam conditions transversal.)

Both steps depend on the following lemma:

Lemma 4.4.1. The map of totally real Grassmannians U(n1)/SO(n1) U(n1+n2)/SO(n1+n2) induces an isomorphism of fundamental groups, and except in the case n1 = 1, an iso-morphism of second homotopy groups.

Proof. By the long exact sequence of homotopy groups and the isomorphisms 1(SO(n1)) 1(SO(n1 + n2)), n1 > 1 and 1(U(n1)) 1(U(n1 + n2)).

Proposition 4.4.2. A relative spin structure onF, deformation of the asymptotic boundaryand seam conditions to split form, and orientations for the ends of each component togetherinduce an orientation on the determinant line bundle det(DE,F) B.

Proof. For simplicity, we assume that the Hamiltonian perturbations on the strip-like endsvanish. We may assume that the ranks of the bundles are at least two, after stabilizing byadding trivial bundles. By the Lemma, there exists a deformation of the seam conditionson the strip-like ends to split form in (U(

nj)/SO(

nj))

2, where n1, . . . , nk are thedimensions of the boundary and seam conditions, such that the path has Maslov index

zero. Any such path has a deformation with no crossing points, that is, so that every setof conditions in the deformation are transversal. This deformation produces an family ofFredholm operators, and hence an isomorphism of determinant line of the original problemwith the problem with split form on each strip-like end. Using the Lemma again, the given


31/44


path can be deformed into a path in partially split form, that is, a path into

U(n1)/SO(n1) U(n1 + n2)/SO(n1 + n2) . . . U(nk)/SO(nk)

uniquely up to homotopy of homotopies. We apply the Lemma a final and third time tohomotope the seam conditions to a set of boundary and seam conditions Fsplit of split formover the entire surface. The index problem on (E, Fsplit) splits into a sum of problems on

the various components, and by the unquilted case we obtain orientations on the variousdeterminant lines. These are then pulled back under the deformations to an orientation onthe determinant line on the original family of operators.

Recall that for a disconnected unquilted surface, the orientation constructed on a Cauchy-Riemann operator depends on an ordering of the components. In particular for a quiltedsurface, the orientation depends on an ordering of the components. However,

Proposition 4.4.3. If S is connected and DE,F has index zero resp. one then the orienta-tion on DE,F is independent of the ordering of the components of S.

Proof. Since the orientation constructed is independent of the choice of deformation to splitform, we may deform F to boundary bundles of split form such that the index is zero oneach resp. all but one patch of S. Then the determinant lines for all component commute,see Proposition 4.2.1.

Finally we discuss the effect of gluing on orientations. In the quilted case, there are fourtypes of gluing to consider: gluing at the interior, gluing on the true boundary, gluing at theseams, and gluing along strip-like ends. These reduce to the corresponding gluing operationson disconnected unquilted surfaces, after deformation of the boundary conditions to splitform. Suppose that DE,F has index zero or one; then we can find a split deformation so thatthe index is one on at most one of the unquilted components. Then the determinant linescorresponding to the various unquilted operators commute. Permuting the components tobe glued adjacent in the ordering and applying the gluing operation for the unquilted caseresults in a collection of operators that again have at most one with index 1, and permutingthe components into the desired ordering does not change the gluing sign. Hence the gluing

sign is the product of gluing signs for the unquilted components. In particular, in the casethat S, S+ are obtained by thickening the boundary of an unquilted surface, and S hasa single outgoing end, this convention leads to a positive sign in the gluing rule, which givesthe associativity relation in the generalized Donaldson-Fukaya category.

5. Orientations for pseudoholomorphic surfaces

5.1. Relative spin structures for sequences of Lagrangians and the toggle functor.Let L = (L0, . . . , Ld) be a sequence of oriented Lagrangian submanifolds in M. A relativespin structure for (L0, . . . , Ld) is a relative spin structure for the immersion L0 . . . Ld M. In particular, this means that each Lj has a relative spin structure with the samebackground class.

Define an involution on the set of relative spin structures on a Lagrangian submanifoldL as follows. The bundle T M has a canonical splitting (up to homotopy) after restrictionto any Lagrangian submanifold L: T M|L = T L T L. It follows from (32) that T M|L hasa canonical spin structure, up to isomorphism.

8/3/2019 Katrin Wehrheim and Chris T. Woodward- Orientations for Pseudoholomorph

Documents

Katrin Wehrheim and Chris T. Woodward- Orientations for Pseudoholomorphic Quilts