Resolving the ambiguities in overall sign

Thus far, we have been content with having an overall sign ambiguity in the ho-momorphisms on Floer homology groups which arise from cobordisms. We now turn to consider what is involved in resolving these ambiguities.

We begin with the case of^BLINK, in which a typical morphism from(Y1,K1,P1) to(Y0,K0,P0)is represented by a cobordism with singular bundle data, (W,S,P). We already have a rather tautological way to deal with the sign issue in this case: we need to enrich our category by including all the data we used to make the sign explicit. Thus, we can define a category^BLINK in which an object is a tuple

(Y0,K0,P0,a0),

where a0 is the auxiliary data consisting of a choice of Riemannian metric ˇg0, a choice of perturbationπ₀, and a choice of basepointθ₀. (See Equation (15) above.) A morphism in^BLINK, from(Y1,K1,P1,a1)to(Y0,K0,P0,a0), consists of the previous data(W,S,P) together with a choice of an I-orientation for (W,S,P) (Definition 3.9). With such a definition, we have a functor (rather than a projective functor)

I:^BLINK →^GROUP.

We can make something a little more concrete out of this for the functor I:^LINK∗ →^P-GROUP.

We would like to construct a category^LINK∗and a functor I:^LINK∗ →^GROUP.

To do this, we first define the objects of^LINK∗to be quadruples(Y,K,x, v), where K is now an oriented link in Y and x andvare a basepoint and normal vector to K as before.

(The orientation is the only additional ingredient here.) Given this data, we can orient the link K by orienting the new component L so that it has linking number 1 with K in the standard ball around the basepoint x. There is a standard cobordism of oriented pairs, (Z,F), from (Y,K)to (Y,K): the 4-manifold Z is a product[0,1] ×Y and the surface F is obtained from the product surface[0,1] ×K by the addition of a standard embedded 1-handle. Alternatively,(Z,F), is a boundary-connect-sum of pairs, with the first summand being[0,1] ×(Y,K)and the second summand being(B⁴,A), where A is a standard oriented annulus in B⁴ bounding the oriented Hopf link in S³. The arcω in Y joining K to L in the direction of vis part of the boundary of a disk ω_Z in Z whose boundary consists ofω⊂Y together with a standard arc lying on F.

Although(Y,K,∅)may not satisfy the non-integral condition, we can nevertheless form the space of connections B(Y,K)=B^∅(Y,K) as before: this is a space of SU(2) connections on the link complement, with holonomy asymptotic to the conjugacy class of the element

We can now exploit that fact that there is (to within some inessential choices) a pre-ferred basepointθ inB(Y,K)arising from a reducible connection, as in Section 3.6 of [21]. Thus, the singular connectionθ is obtained from the trivial product connection in SU(2)×Y by adding a standard singular term

β(r)i 4η

whereβis a cut-off function on a tubular neighborhood of K andηis (as before) a global angular 1-form, constructed this time using the given orientation of K. If β is a critical point for the perturbed functional inB^ω(Y,K), we let Aθ,β be any chosen connection in B^ωZ(Z,F;θ, β) (i.e. a connection on the cobordism, asymptotic to θ andβ on the two ends). We then define

(25) (β)

to be the two-element set of orientations for the determinant line det(DAθ,β).

To summarize, we have defined (β) much as we defined(β) earlier in (12).

The differences are that we are now using the non-trivial cobordism (Z,F, ωZ) rather than the product, and we are exploiting the presence of a distinguished reducible con-nection on the other end of this cobordism, defined using the given orientation of K. We can define a chain complex

C(Y,K)=

β

Z(β),

and we can regard I(Y,K)as being defined by the homology of this chain complex.

Consider next a morphism in ^LINK∗, say (W,S, γ , v) from (Y1,K1,x1, v₁) to (Y0,K0,x0, v₀). We shall choose orientations for K1 and K0, as we did in the previous paragraphs. We do not assume that the surface S is an oriented cobordism, but we do require that it looks like one in a neighborhood of the pathγ: that is, we assume that if ui is an oriented tangent vector to Ki at xi, then there is an oriented tangent vector to S alongγ, normal toγ, which restricts to u1and u0at the two ends.

We have an associated morphism (W,S, ω) between (Y1,K₁, ω₁) and (Y0,K₀, ω₀). Let

β₁∈B^ω1(Y1,K1) β₀∈B^ω0(Y0,K0)

be critical points, let[Aβ1,β0]be a connection inB^ω(W,S;β₁, β₀), and consider the prob-lem of orienting the determinant line

det(DAβ1,β0).

Let(Z1,F1)and(Z0,F0)be the standard cobordisms, described above, (Zi,Fi):(Yi,Ki)→(Yi,K_i).

There is an evident diffeomorphism, between two different composite cobordisms, from (Y1,K1)to(Y0,K₀):

(Z1,F1, ω_Z1)∪Y1(W,S, ω)=(W,S,∅)∪Y0 (Z0,F0,Y0).

From this we obtain an isomorphism of determinant lines,

(26) det(DA_θ1,β1)⊗det(DA_β1,β0)=det(DA_θ1,θ0)⊗det(DA_θ0,β0).

Here θ_i are the preferred reducible connections in B(Yi,Ki) as above, and Aθ1,θ0 is a connection joining them across the cobordism(W,S).

If we wish the cobordism(W,S, ω)to give rise to a chain map C(Y1,K1)→C(Y0,K0)

with a well-defined overall sign, then we need to specify an isomorphism det(DA_β1,β0)→Hom

Z(β1), (Z(β0)

;

and by the definition ofand the isomorphism (26), this means that we must orient det(DAθ1,θ0).

Thus we are led to the following definition:

Definition4.3. — Let(Y1,K1)and(Y0,K0)be two pairs of oriented links in closed oriented 3-manifolds, and let(W,S)be a cobordism of pairs, from(Y1,K1)to(Y0,K0), with W an oriented cobordism, and S and unoriented cobordism (and possible non-orientable). Then an I-orientation for (W,S)will mean an orientation for the determinant line det(DAθ1,θ0), whereθ_iare the reducible singular SU(2)connections on Yi, described above and determined by the given orientations of the Ki.

In the special case that W is a product[0,1] ×Y, an I-orientation for an embedded cobordism S between oriented links will mean an I-orientation for the pair([0,1] ×Y,S).

We are now in a position to define^LINK∗. Its objects are quadruples(Yi,Ki,xi, v_i) with Kian oriented link, and its morphisms are quintuples(W,S, γ , v, λ), where

– (W,S)is a cobordism of pairs, with W an oriented cobordism;

– γ is a path from x1 to x0, with the property that S is an oriented cobordism alongγ;

– vis a normal vector to S alongγ, restricting tov₁,v₀at the two ends; and – λis a choice of I-orientation for(W,S).

With this definition, we have a well-defined functor to groups.

Remarks. — Our definition of I-orientation still rests on an analytic index, so some comments are in order. First of all, the definition makes it apparent that there is a natural composition law for I-orientations of composite cobordisms. Second, if S is actually an oriented cobordism from K1to K0, then an I-orientation of(W,S)becomes equivalent to a homology-orientation of the cobordism, as discussed in [20] and [21]. Indeed, the case that S is oriented is precisely the case considered in [21], where homology-orientations of the cobordisms are shown to fix the signs of the corresponding chain-maps. In particular, if W is a product[0,1] ×Y, then an oriented cobordism S between oriented links in Y has a canonical I-orientation, and these canonical I-orientations are preserved under composition.

When using the unreduced functor I for knots inR³, we have adopted the con-vention of putting the extra Hopf link “at infinity” in a standard position. With this setup, we have a category

(27) ^LINK(R³)

whose objects are oriented links inR³and whose morphisms are I-oriented cobordisms in[0,1] ×R³.