2.1 Instanton homology with constant coefficients

A trivalent graph, or web, $K$ in a closed oriented $3$-manifold $Y$ gives rise to an orbifold, which we will simply denote by $(Y,K)$. The isotropy groups are taken to be $\mathbb{Z}/2$ along edges of $K$ and $\mathbb{Z}/2 \times \mathbb{Z}/2$ at the vertices. We refer to such a special orbifold as a bifold and we consider orbifold $\mathop {{SO}}(3)$ bundles (or bifold bundles) $E$ over $(Y,K)$ requiring that the local isotropy groups of the orbifold act effectively on the $\mathop {{SO}}(3)$ fibers. Marking data on $(Y,K)$ consists of an open set $U_{\mu }$ and a bifold bundle $E_{\mu }\to U_{\mu } {\setminus} K$; a bifold connection is marked by $\mu$ if an isomorphism $\sigma : E_{\mu } \to E|_{U_{\mu }}$ is given. An isomorphism $\tau$ between $\mu$-marked bundles with connection, $(E,A,\sigma )$, $(E',A',\sigma ')$, is an isomorphism of bifold bundles-with-connection such that the automorphism $\sigma ^{-1}\tau \sigma ' : E_{\mu }\to E_{\mu }$ lifts to the determinant-1 gauge group. We write

\[ \mathcal{B}(Y, K ; \mu) \]

for the space of isomorphism classes of $\mu$-marked bifold bundles with connection.

The marking data $\mu$ is strong if the automorphism group of every flat $\mu$-marked bifold connection is trivial. A sufficient condition is that $U_{\mu }$ contains a vertex of $K$ and in this case there are indeed no connections with non-trivial stabilizer even in $\mathcal {B}(Y, K; \mu )$. With coefficients in the field $\mathbb{F}_{2}$ of two elements, one can construct an $\mathop {{SO}}(3)$ instanton homology group

\[ J((Y,K) ; \mu) \]

for any bifold with strong marking data. The generators of the complex from which this instanton homology is computed correspond to critical points of a perturbed Chern–Simons functional on $\mathcal {B}(Y, K ; \mu )$. We may omit $Y$ from our notation for both $J$ and $\mathcal {B}$ when $Y$ is understood (which is often the case when $Y$ is $S^{3}$).

Consider next a framed base point $y_{0} \in Y$ with standard neighborhood $B(y_{0}) \cong B^{3}$. We write $Y^{o}$ for the complement of this standard neighborhood:

\[ Y^{o} = Y {\setminus} B(y_{0}). \]

Given a web

\[ K \subset Y^{o}, \]

we may form a new web as a split union

(13)\begin{equation} K^{\sharp} = K \cup \theta, \end{equation}

where $\theta \subset B(y_{0})$ is a standard theta graph (three edges and two vertices) contained in the ball. We may then define

(14)\begin{equation} J^{\sharp}(K) = J((Y, K^{\sharp}); \mu_{\theta}), \end{equation}

where the marking data $\mu _{\theta }$ consists of the ball $U_{\theta } = B(y_{0})$ containing $\theta$, with $E_{\theta }$ the unique trivial bundle on $B^{3}{\setminus} \theta$. The group $J^{\sharp }(K)$ was defined first in [Reference Kronheimer and MrowkaKM19b], though the description in that paper was a slight variant of this one. The description we have just given is from [Reference Kronheimer and MrowkaKM19a], where the equivalence of the two descriptions is also proved.

In this paper, we will be almost exclusively concerned with the case that the marking region $U_{\mu }$ is not just the ball $B(y_{0})$ but is instead the whole of $Y$. The distinguished $\mathop {{SO}}(3)$ bundle $E_{\mu }$ on $Y{\setminus} K^{\sharp }$ may in general have non-zero Stiefel–Whitney class

\[ w_{2}(E_{\mu}) \in H^{2} ( Y{\setminus} K^{\sharp} ; \mathbb{Z}/2). \]

We take this class to be represented by $\omega \subset Y$, which is a codimension-2 submanifold with boundary. We make the following assumptions on $\omega$:

• – $\omega$ is (the interior of) a union of circles and arcs with end points on $K$;

• – $\omega$ is disjoint from the ball $B(y_{0})$ which contains $\theta$.

We require that $\omega$ represents $w_{2}(E_{\mu })$, in the sense that

\[ w_{2}(E_{\mu}) = \mathop{PD}\big[\omega \cap (Y{\setminus} K^{\sharp})\big]. \]

Having chosen $\omega$, we shall trivialize $E_{\mu }$ on the complement of $\omega$, so that the obstruction to extending the trivialization across each component of $\omega$ is non-zero. We use this trivialization to give a lift of $E_{\mu }$ to an $\mathop {{SU}}\nolimits (2)$ bundle on the complement of $\omega$.

Let us write $\mu _{\omega }$ for the marking data obtained in this way from a 1-manifold $\omega \subset Y {\setminus} K^{\sharp }$.

Definition 2.1 Using the marking data $E_{\mu }$ as above, whose Stiefel–Whitney class is dual to $\omega \subset Y{\setminus} K$, we write

(15)\begin{equation} I^{\sharp}(K)_{\omega} = J((Y, K^{\sharp}); \mu_{\omega}) \end{equation}

for the corresponding instanton homology of the web $K\subset Y^{o}$. We also write

(16)\begin{equation} \mathcal{B}^{\sharp}(K)_{\omega} = \mathcal{B}\big((Y, K^{\sharp}) ; \mu_{\omega}\big) \end{equation}

for the corresponding configuration space of connections. When $\omega$ is empty, we simply omit it from our notation and write $I^{\sharp }(K)$.

When $K$ is a knot or link, this variant coincides with the mod-2 version of the invariant $I^{\sharp }(K)_{\omega }$ as introduced in [Reference Kronheimer and MrowkaKM12] (though in that paper the coefficient ring was $\mathbb{Z}$). As with $J^{\sharp }$, the definition we have presented here is slightly different from the earlier one: the difference is the use of the graph $\theta$ in place of the Hopf link that was used in [Reference Kronheimer and MrowkaKM12]. But the two definitions give isomorphic homology groups, by the arguments from [Reference Kronheimer and MrowkaKM19a].

Because the marking data is all of $Y$, the gauge theory which underlies this instanton homology is essentially an $\mathop {{SU}}\nolimits (2)$ gauge theory. In particular, we have the following identification.

2.2 The local system

We begin with some motivation of our construction. If $\pi$ denotes the fundamental group of the configuration space $\mathcal {B}^{\sharp }(K)_{\omega }$, then there is a tautological local system over $\mathcal {B}^{\sharp }(K)_{\omega }$ whose fiber at each point is a free rank-1 module for the group ring $\mathbb{F}_{2}[\pi ]$. It can be realized by defining its fiber at $[A] \in \mathcal {B}^{\sharp }(K)_{\omega }$ to be the vector space of $\mathbb{F}_{2}$-valued functions with finite support on the fiber of the universal cover $\tilde {\mathcal {B}}^{\sharp }(K)_{\omega } \to \mathcal {B}^{\sharp }(K)_{\omega }$. Given any choice of homomorphism

\[ \epsilon: \pi \to G, \]

there is a corresponding local system $\Gamma _{\epsilon }$ of $\mathbb{F}_{2}[G]$ modules. Our instanton homology is $\mathbb{Z}/2$ graded, not $\mathbb{Z}$ graded, because of non-trivial spectral flow, and there is an infinite cyclic cover of $\mathcal {B}^{\sharp }(K)$ on which the spectral flow is trivial. Let

\[ \pi' \subset \pi \]

be the fundamental group of this infinite cyclic cover. The instanton homology groups are essentially unchanged in passing to the cover (the homology becomes $\mathbb{Z}$ graded and 2-periodic, rather than $\mathbb{Z}/2$ graded). Up to isomorphism, the instanton homology groups with coefficients in the local system $\Gamma _{\epsilon }$ will therefore depend on the homomorphism $\epsilon$ only through the restriction of $\epsilon$ to $\pi '$.

Although we shall not need a proof, the fundamental group $\pi$ is a free abelian group of rank $5$ when $K$ is a knot and $\omega$ is empty. It follows in this case that $\pi '$ has rank $4$ and we will therefore capture the most general local system if we construct a homomorphism

\[ \pi \to \mathbb{Z}^{4} \]

which is injective on the subgroup $\pi ' \subset \pi$.

The construction described in [Reference Kronheimer and MrowkaKM19a] arises from a map

\[ \pi \to \mathbb{Z}^{3} \]

presented in terms of an explicit map

\[ \big(h_{1}, h_{2}, h_{3}\big): \mathcal{B}^{\sharp}(K)_{\omega} \to S^{1} \times S^{1} \times S^{1} \]

and this leads to the local system $\Gamma _{\theta }$ of free rank-1 modules over $\mathbb{F}_{2}[\mathbb{Z}^{3}]$ as described at (2) in the introduction. To recall this briefly from [Reference Kronheimer and MrowkaKM19a], the marking data $\mu _{\omega }$ means that our gauge theory has structure group $\mathop {{SU}}\nolimits (2)$ and, at the two vertices of $\theta$, the structure group of the two fibers $E_{+}$ and $E_{-}$ is reduced to the center $\{\pm 1\}$. Along each edge of $\theta$, the structure group is reduced to $S^{1}$. The holonomy along each edge therefore gives a well-defined element of

(17)\begin{equation} S^{1} / \{{\pm} 1\} \cong \mathbb{R}/\mathbb{Z}. \end{equation}

Applied to the three edges of $\theta$ in turn, one obtains the three components $h_{1}$, $h_{2}$, $h_{3}$. (The notation for $\Gamma _{\theta }$ was simply $\Gamma$ in [Reference Kronheimer and MrowkaKM19a].)

When $\omega$ has no end points (so is disjoint from $K$), a very similar construction from [Reference Kronheimer and MrowkaKM13] can be adapted to define a map

(18)\begin{equation} h_{0} : \mathcal{B}^{\sharp}(K)_{\omega} \to S^{1}. \end{equation}

To describe this, consider first the case that $K$ is a knot. Choose a framing $\tau$ for $K$ so as to have well-defined push-off. As explained in [Reference Kronheimer and MrowkaKM11], the framing allows us to interpret the orbifold connection $[A]$ as giving rise to a well-defined connection over the knot $K$ itself, carried by a bundle with an involution $g$ coming from the orbifold structure. Because of the action of $g$, the adjoint bundle decomposes as a sum

\[ \xi \oplus \eta, \]

where $\xi$ is a real line bundle on $K$ and $\eta$ is a 2-plane bundle. The marking data allows us to identify the orientation bundle of the knot $K$ with the orientation bundles of both $\xi$ and $\eta$, so the connection in the 2-plane bundle $\eta$ has a well-defined circle-valued holonomy along $K$. The holonomy of $\eta$ around $K$ is the definition of $h_{0}$ above. If $K$ is a link rather than a knot, we multiply the holonomies along all the components.

Combining the two previous constructions, we now have a map

(19)\begin{equation} \big(h_{0}, h_{1}, h_{2}, h_{3}\big) : \mathcal{B}^{\sharp}(K)_{\omega} \to \mathbb{R}^{4}/\mathbb{Z}^{4} \end{equation}

whenever $\omega$ has no boundary points. In the case that $\omega$ has boundary on $K$, the component $h_{0}$ must be omitted. As in [Reference Kronheimer and MrowkaKM11], we use an explicit description of the corresponding local system that depends on the maps $h_{i}$ but does not depend on a choice of base point in $\mathcal {B}^{\sharp }(K)_{\omega }$. We write

\[ \mathcal{R} = \mathbb{F}_{2}[ \mathbb{Z}^{4}] \]

and regard this as a subring of $\mathbb{F}_{2}[\mathbb{R}^{4}]$. For each $\mu \in \mathbb{R}^{4}$, we have the rank-1 $\mathcal {R}$-module

\begin{align*} \Gamma|_{\mu} &= T_{0}^{\mu_{0}} T_{1}^{\mu_{1}} T_{2}^{\mu_{2}} T_{3}^{\mu_{3}} \mathcal{R} \\ &\subset \mathbb{F}[\mathbb{R}^{4}] \end{align*}

and these form a local system over the torus $\mathbb{R}^{4}/\mathbb{Z}^{4}$. Pulling this back by the map (19), we obtain our local system $\Gamma$ over $\mathcal {B}^{\sharp }(K)_{\omega }$.

We summarize these constructions as follows.

Notation 2.3 Let $K \subset Y^{o}$ be a link, let $K^{\sharp } = K \cup \theta$ and let $\omega \subset Y$ be a 1-manifold without boundary, disjoint from the ball containing $\theta$. Let $\mathcal {B}^{\sharp }(K)_{\omega }$ be the associated space of connections. On $\mathcal {B}^{\sharp }(K)_{\omega }$, we have a local system $\Gamma$ of (free, rank-1) modules over

\[ \mathcal{R}=\mathbb{F}_{2}[\mathbb{Z}^{4}]. \]

For any base change $\sigma :\mathcal {R}\to \mathcal {S}$, there is a corresponding system of $\mathcal {S}$-modules,

\[ \Gamma_{\sigma} = \Gamma\otimes_{\sigma}\mathcal{S}. \]

If $\sigma (T_{0})=1$, then the map $h_{0} : \mathcal {B}^{\sharp }(K)_{\omega } \to S^{1}$ is not required in the construction of the local system, and in this case we can form the local system $\Gamma _{\sigma }$ more generally, when $\omega$ is allowed to be a manifold with boundary with end points on $K$.

Remarks (i) In the case that $\omega$ has boundary and $\sigma (T_{0})=1$, our notation for the local system $\Gamma _{\sigma }$ involves a slight abuse of notation, since we can no longer write it as $\Gamma \otimes _{\sigma } \mathcal {S}$. (The local system $\Gamma$ is not defined in this case.) It should more properly be defined as $\Gamma _{\theta }\otimes _{\bar {\sigma }} \mathcal {S}$, where $\bar {\sigma } : \mathbb{F}_{2}[\mathbb{Z}^{3}] \to \mathcal {S}$ is the map through which $\sigma$ factors.

(ii) The definition of $h_{0}$ above makes use of a framing $\tau$ for the knot (or for each component of the link). If the framing $\tau$ is changed by $1$, then $h_{0}$ changes to $h_{0} + 1/2$. (See [Reference Kronheimer and MrowkaKM11].) Therefore, framings whose difference is even give rise to the same map $h_{0}$ and identical local systems $\Gamma$.

(iii) The four maps $h_{i}$ in (19) give a map

\[ \phi: \pi = \pi_{1}(\mathcal{B}^{\sharp}(K)_{\omega}) \to \mathbb{Z}^{4} \]

whenever $\omega$ has no boundary points. We can say a little more about the kernel and image of $\phi$. The space $\mathcal {B}^{\sharp }(K)_{\omega }$ is connected and can be identified as usual with a quotient $\mathcal {A} / \mathcal {G}$ of an affine space of connections by the action of the gauge group. We can therefore identify $\pi$ as $\pi _{0}(\mathcal {G})$. When $K$ is a knot, the kernel of the map $\phi$ is $\mathbb{Z}$ and consists of the components of $\mathcal {G}$ represented by gauge transformations that are supported in the neighborhood of a point in $S^{3}{\setminus} K$. The image of $\phi$ is a sublattice $\Lambda \subset \mathbb{Z}^{4}$ of index $8$. In terms of the standard basis $v_{i}$, this lattice is generated by the elements

\[ 2v_{i}\, (i=0,1,2,3) \quad \text{and} \quad v_{1} + v_{2} + v_{3}. \]

For example, the fact that the $v_{0}$ coefficient is even is a reflection of the fact that the map $h_{0}$ lifts to the double cover of $S^{1}$. In turn this lift exists because we can use the holonomy of the $\mathop {{SU}}\nolimits (2)$ connection around the loop $K$ to define a map $\tilde {h}_{0}$, rather than use the holonomy of the $\mathop {{SO}}(3)$ connection, which defines $h_{0}$. In the same way, the fact that the coefficients of $v_{1}$ and $v_{2}$ have the same parity similarly means that $h_{1}+h_{2}$ lifts to a double cover, essentially because the corresponding pair of edges of $\theta$ form a closed loop. Instead of the ring $\mathbb{F}_{2}[\mathbb{Z}^{4}]$, we could instead work with the subring $\mathbb{F}_{2}[\Lambda ]$, which we can identify as the subring generated by the monomials $T_{i}^{\pm 2}$ and $T_{1}T_{2}T_{3}$.

(iv) The previous remark explains that the circle-valued map $h_{0}$ has a lift $\tilde {h}_{0}$, through the double cover of $S^{1}$, and this accounts for a difference in conventions between the present paper and (for example) the notation in [Reference Kronheimer and MrowkaKM13]. In [Reference Kronheimer and MrowkaKM13], the local system is defined using the map $\tilde {h}_{0}$ and is described as a module for the ring of finite Laurent series in a formal variable $u$. Because of the double cover, the variable $u$ in that paper corresponds to $T_{0}^{2}$ in the present paper, rather than $T_{0}$. Our present choice of conventions is for consistency with [Reference Kronheimer and MrowkaKM19a].

(v) In the case that $K$ is an $n$-component link and $\omega$ has no boundary points, the fundamental group $\pi$ is free of rank $4+n$ and $\pi '$ is free of rank $3+n$. If $\omega$ has boundary, and if $k$ is the number of components of $K$ on which $\partial \omega$ has an odd number of points, then the rank of $\pi '$ is $3+n-k$ and the torsion subgroup of $\pi '$ is $(\mathbb{Z}/2)^{k-1}$. The proofs of these assertions are essentially the same as the results of § 3.2 of [Reference Kronheimer and MrowkaKM12].

2.3 The chain complex

Following Notation 2.3 henceforth, we fix a base-change homomorphism $\sigma :\mathcal {R}\to \mathcal {S}$, possibly the identity. We fix a 3-manifold $Y$, a base point $y_{o}\in Y$, a link $K \subset Y$ disjoint from a fixed ball around the base point and a representative $\omega$ for the Stiefel–Whitney class. If $\sigma (T_{0})=1$, then we allow $\omega$ to have boundary on $K$.

We can now construct the chain complex and boundary map which will define a Floer homology group $I^{\sharp }(K;\Gamma _{\sigma })_{\omega }$ in the usual way for an instanton Floer homology. While the construction is a straightforward generalization of the treatment in [Reference Kronheimer and MrowkaKM19a] and its predecessors, it is worthwhile to recall a particular point from [Reference Kronheimer and MrowkaKM19a], namely the proof that $\partial ^{2}=0$ given in [Reference Kronheimer and MrowkaKM19a, Lemma 3.1]. From there we see that, a priori, there is a relation of the form

\[ \partial \circ \partial = W \mathbb{1} \]

for some $W\in \mathcal {S}$. That is, we may have a ‘matrix factorization’ rather than a complex. The proof that $W=0$ carries over from [Reference Kronheimer and MrowkaKM19a] without change: it vanishes because it is a sum of contributions from the vertices of $\theta$ and is independent of $K$. Although we will not pursue this further in the present paper, it is worth observing what happens in a more general situation. Suppose that we consider the case that $K$ is a web rather than a link and suppose that we use each edge $e$ of $K$ to define a map $h_{e} : \mathcal {B}^{\sharp }(K)\to S^{1}$, so as to obtain a local system of modules over a ring of Laurent series in variables $T_{e}$ indexed by the edges. The term $W$ will have contributions from possible bubbling at the vertices $v$ of $K$, so

\[ W = \sum_{v} W_{v}. \]

With a little more care, one may explicitly compute $W_{v}$ and it has the form

\[ W_{v} = p\big( T_{e(v,1)}, T_{e(v,2)}, T_{e(v,3)}\big), \]

where $e(v,i)$ are the three edges incident at $v$ and

\[ p(T_{1}, T_{2}, T_{3}) = T_{1}T_{2}T_{3} + T_{1}T_{2}^{{-}1}T_{3}^{{-}1} + T_{2}T_{3}^{{-}1}T_{1}^{{-}1} + T_{3}T_{1}^{{-}1}T_{2}^{{-}1} \]

is the same polynomial that defines $P$. (Our notation here as elsewhere will sometimes not distinguish a generator $T_{i}\in \mathcal {R}$ from its image under the base change, $\sigma (T_{i})\in \mathcal {S}$.) In this generality, the potential $W$ is non-zero. It becomes zero if we impose relations on the variables $T_{e}$ so as to ensure (for example) that the product

\[ T_{e(v,1)} T_{e(v,2)} T_{e(v,3)} \]

is independent of the vertex $v$.

2.4 Functoriality

Let $X$ be an oriented four-dimensional cobordism from $Y_{0}$ to $Y_{1}$ and let $S\subset X$ be a surface (not necessarily orientable) which provides a cobordism between links $K_{0}\subset Y_{0}$ and $K_{1}\subset Y_{1}$. Because of the need for a base point, we suppose that $X$ contains an embedded cylinder $[0,1]\times B^{3}$ whose boundary at the two ends are the balls $B(y_{0})$ and $B(y_{1})$ around the chosen base points. We suppose that $S$ is disjoint from this cylinder, so we may form the larger foam

\begin{align*} S^{\sharp} &= S \cup \big([0,1]\times \theta \big)\\ & \subset X. \end{align*}

The foam $S^{\sharp } \subset X$ provides an orbifold structure on $X$, which we write as $(X,S^{\sharp })$.

The oriented orbifold $(X,S^{\sharp })$ is a cobordism between the orbifolds $(Y_{i}, K_{i}^{\sharp })$. As a special case of the general machinery of [Reference Kronheimer and MrowkaKM19b] and [Reference Kronheimer and MrowkaKM12], it defines homomorphisms on the constant-coefficient instanton homology groups

\[ I^{\sharp}(X,S) : I^{\sharp}( Y_{0}, K_{0}) \to I^{\sharp}(Y_{1}, K_{1}). \]

More generally, we can again allow bundles with non-zero $w_{2}$ represented as the dual of a submanifold $\omega$. As in [Reference Kronheimer and MrowkaKM12], we take $\omega$ to be a surface with corners. Thus, the boundary of $\omega$ consists of:

• – a 1-manifold $\omega _{0}\subset Y_{0}$, possibly with boundary on $K_{0}$;

• – a 1-manifold $\omega _{1}\subset Y_{1}$ similarly; and

• – a union of arcs and circles in the surface $S\subset Y$.

As well as meeting $S$ along its boundary, we allow $\omega$ to meet $S$ also in its interior, in transverse points of intersection [Reference Kronheimer and MrowkaKM12]. We then have the more general functoriality, with maps

\[ I^{\sharp}(X,S)_{\omega} : I^{\sharp}( Y_{0}, K_{0})_{\omega_{0}} \to I^{\sharp}(Y_{1}, K_{1})_{\omega_{1}}. \]

We can now introduce the local system $\Gamma _{\sigma }= \Gamma \otimes _{\sigma }\mathcal {S}$. If $\partial \omega$ meets $S$, then we require that $\sigma (T_{0})=1$, as in Notation 2.3. When the local system is introduced and $\sigma (T_{0})\ne 1$, we must also take additional care, because of the role of the framings. Recall that the map $h_{0} : \mathcal {B}^{\sharp }(K) \to S^{1}$ depends on a choice of framing of $K$ and otherwise has an ambiguity of a half-period. Framings which have different parities give rise to groups $I^{\sharp }(K ; \Gamma _{\sigma })_{\omega }$ that are isomorphic, but not canonically so without further choices. This issue is dealt with carefully in [Reference Kronheimer and MrowkaKM11] (and with some inessential inaccuracies in [Reference Kronheimer and MrowkaKM13]). We recall the procedure.

Let us recall first that the construction of $I^{\sharp }(K;\Gamma _{\sigma })_{\omega }$ depends on framing of $K$, and that the map that we are seeking to define should therefore be written

(20)\begin{equation} I^{\sharp}(X,S ; \Gamma_{\sigma})_{\omega} : I^{\sharp}( Y_{0}, K_{0}; \Gamma_{\sigma})^{\tau_{0}}_{\omega_{0}} \to I^{\sharp}(Y_{1}, K_{1}; \Gamma_{\sigma})^{\tau_{1}}_{\omega_{1}}, \end{equation}

where we have now included the framings of $K_{0}$ and $K_{1}$ explicitly in the notation. At the chain level, the map will be given by a chain map

\[ C^{\sharp}(S; \Gamma_{\sigma})_{\omega} : C^{\sharp}( Y_{0}, K_{0}; \Gamma_{\sigma})^{\tau_{0}}_{\omega_{0}} \to C^{\sharp}(Y_{1}, K_{1}; \Gamma_{\sigma})^{\tau_{1}}_{\omega_{1}} \]

whose matrix entry from $\alpha _{0}$ to $\alpha _{1}$ is given by ‘counting instantons’ as usual, and attaching a ‘weight’ $\epsilon ([A]) \in \mathcal {R}$ to each instanton $[A]$ from $\alpha _{0}$ to $\alpha _{1}$.

To define $\epsilon ([A])$, following [Reference Kronheimer and MrowkaKM11], we note that the framings of $K_{0}$ and $K_{1}$ allow one to define a self-intersection number $S\cdot S$ for the surface $S$. Given the instanton $[A]$ on the cylindrical-end orbifold obtained from $(X,S)$, one may define locally an $\mathop {{SO}}(3)$ bundle on $S$ with a reducible connection, so that the associated $\mathbb{R}^{3}$ bundle has locally the form $\xi \oplus \eta$ (a line bundle and a 2-plane bundle). Although the construction is local, the curvature $2$-form of the 2-plane bundle $\eta$ exists globally on $S$ as a 2-form $\Omega$ with values in the orientation bundle of $S$. So, we may consider the integral

\[ \nu_{0}(A) = c \int_{S} \Omega \]

with the normalizing constant $c$ chosen so that $\nu _{0}$ coincides with the Euler class of $\eta$ in the closed orientable case. (So $c= i/2\pi$ if we identify the Lie algebra of the circle with $i\mathbb{R}$ in the usual way.) An application of Stokes’ theorem gives

(21)\begin{equation} \nu_{0}(A) + (1/2)(S\cdot S) = h_{0}(\alpha_{1}) - h_{0}(\alpha_{0}) \quad ({\rm mod} \ \mathbb{Z}). \end{equation}

We may define similar quantities $\nu _{i}(A)$ ($i=1,2,3$) as the integrals of the curvature on the three edges of $\theta$. The weight $\epsilon ([A])$ is now defined by

(22)\begin{equation} \epsilon([A]) = T_{0}^{\nu_{0}(A) + (1/2)(S\cdot S)} T_{1}^{\nu_{1}(A)} T_{2}^{\nu_{2}(A)} T_{3}^{\nu_{3}(A)}. \end{equation}

These relations (21), and the simpler formulae for the other $\nu _{i}$, mean that multiplication by (22) is a map from the fiber $\Gamma _{s,\alpha _{0}}$ to $\Gamma _{s,\alpha _{1}}$, as required. The formula for the exponent of $T_{0}$ is the same as in [Reference Kronheimer and MrowkaKM11], except for a factor of $2$ which stems from the difference between $\mathop {{SU}}\nolimits (2)$ and $\mathop {{SO}}(3)$, as explained in Remark (ii) at the end of § 2.2 above.

The result of this construction is a well-defined map (20) between instanton homology groups. As a special case, we may take $K_{0}=K_{1}=K$ and use the cylindrical cobordism to obtain canonical isomorphisms

\[ I^{\sharp}( Y, K; \Gamma_{\sigma})^{\tau_{0}}_{\omega} \to I^{\sharp}(Y, K; \Gamma_{\sigma})^{\tau_{1}}_{\omega}, \]

where only the framing has changed. We use these canonical isomorphisms to treat $I^{\sharp }(Y, K; \Gamma _{\sigma })^{\tau }_{\omega }$ as being independent of the choice of framing $\tau$. Note, however, that if $\tau _{0}$ and $\tau _{1}$ are framings which are equal mod $2$, then the corresponding local systems are identical, but our chosen canonical isomorphism is not the identity map: it is multiplication by $T_{0}^{n}$, where $n$ is half the difference between the framings.

Henceforth, we will continue to omit $\tau$ from our notation. When the ambient cobordism $X$ is a cylinder, or is otherwise understood, we will simply write

(23)\begin{equation} I^{\sharp}(S ; \Gamma_{\sigma})_{\omega} : I^{\sharp}( K_{0}; \Gamma_{\sigma})_{\omega_{0}} \to I^{\sharp}( K_{1}; \Gamma_{\sigma})_{\omega} \end{equation}

for the map (20).

2.5 A reduced variant

The instanton homology $I^{\sharp }(K)$ with constant coefficients has a ‘reduced’ version $I^{\natural }(K)$, which is described, for example, in [Reference Kronheimer and MrowkaKM12]. It depends on a choice of base point on the link $K$, and the relationship of $I^{\natural }$ to $I^{\sharp }$ is similar to the relationship between the reduced and unreduced versions of Khovanov homology [Reference KhovanovKho00]. Given a base change $\sigma : \mathcal {R}\to \mathcal {S}$ satisfying the extra condition $\sigma (T_{0})=\sigma (T_{1})$, we can construct a reduced version $I^{\natural }(Y,K;\Gamma _{\sigma })_{\omega }$ of $I^{\sharp }(Y,K;\Gamma _{\sigma })_{\omega }$, for knots and links $K\subset Y$. We describe the construction here.

Let $K\subset Y$ be a link. Let $p$ be the base point on $K$ and $(\epsilon _{1}, \epsilon _{2}, \epsilon _{3})$ be an oriented basis of tangent vectors with $\epsilon _{1}$ pointing along $K$. Making the modification in a standard ball around $p$, create a spatial graph with two vertices by replacing an arc of $K$ adjacent to $p$ with a bigon, as shown in Figure 1. Let $K^{\natural }\subset Y$ denote the resulting web. Let $\mathcal {B}^{\natural }(Y, K)$, or just $\mathcal {B}^{\natural }(K)$, denote the space of marked bifold connections on the corresponding bifold.

Figure 1.

A knot $K$ with base point $p$ and the resulting web $K^{\natural }$ obtained by adding a bigon.

We define three circle-valued functions,

\[ \big(h_{1}, h_{2}, h_{3}\big) : \mathcal{B}^{\natural}(K) \to \mathbb{R}^{3}/\mathbb{Z}^{3}, \]

as follows. First, in the case that $K$ is a knot, the web $K^{\natural }$ is the union of three oriented arcs $e_{1}$, $e_{2}$, $e_{3}$, where $e_{2}$ and $e_{3}$ comprise the added bigon. All three are oriented by $\epsilon _{1}$. As before, the structure group of a connection $[A]\in \mathcal {B}^{\natural }(K)$ reduces to $S^{1}$ along the arcs, and the holonomy of $[A]$ along the three arcs defines the maps $h_{i}$, just as in the case of $\mathcal {B}^{\sharp }(K)$. If $K$ is a link, let

\[ h_{0} : \mathcal{B}^{\natural}(K) \to \mathbb{R}/\mathbb{Z} \]

be obtained from the holonomy along the remaining components of $K$ (those that do not contain $p$). In the case of a knot, just take $h_{0}$ to be constant. In either case, we now have a map

\[ \big(h_{0},h_{1}, h_{2}, h_{3}\big) \to \mathbb{R}^{4}/\mathbb{Z}^{4}, \]

from which we can construct a local system of $\mathcal {R}$-modules $\Gamma$ over $\mathcal {B}^{\natural }(K)$ as before. In the case that the base change $\sigma : \mathcal {R}\to \mathcal {S}$ has $\sigma (T_{0}) = \sigma (T_{1})$, the local system is pulled back from $\mathbb{R}^{3}/ \mathbb{Z}^{3}$ via the map

\[ \mathbb{R}^{4}/\mathbb{Z}^{4} \to \mathbb{R}^{3}/\mathbb{Z}^{3}, \]

which adds the first two components. We shall consider only cases such as this when discussing reduced instanton homology in this paper, in order to have the components of $K$ on an equal footing. We then define $I^{\natural }(K; \Gamma _{\sigma })$ using the Morse homology of the perturbed Chern–Simons functional on $\mathcal {B}^{\natural }(K)$, with coefficients in $\Gamma _{\sigma }$.

This reduced instanton homology is functorial for ‘based’ cobordisms of links. Given links $(Y_{0}, K_{0})$ and $(Y_{1}, K_{1})$, with framed base points $p_{0}$ and $p_{1}$ on the links, the appropriate morphism is given by a cobordism of pairs, $(X,S)$, together with an arc $\gamma \subset S$ joining the base points and a framing $(\epsilon _{1}, \epsilon _{2}, \epsilon _{3})$ of the normal to $\gamma$ in $X$ such that $\epsilon _{1}$ is tangent to $S$. Equivalently, we can think of an embedding of $[0,1]\times B^{3}$ in $X$ which intersects $S$ in the image of the standard $[0,1]\times B^{1}$. Given such data, we can perform the bigon addition (Figure 1) in a one-parameter family along the image of $[0,1]\times B^{3}$, to obtain an embedded foam $S^{\natural }$ with boundary $K^{\natural }_{0} \cup K^{\natural }_{1}$. The foam gives rise to homomorphisms

\[ I^{\natural}\big(X,S; \Gamma_{\sigma}\big) : I^{\natural}\big(Y_{0}, K_{0}; \Gamma_{\sigma}\big) \to I^{\natural}\big(Y_{1}, K_{1}; \Gamma_{\sigma}\big), \]

where the matrix entries at the chain level are given by the same formulae (22) as in the non-reduced case, with the $\nu _{i}$ being the curvature integrals over the facets of $S^{\natural }$.

The naming of $I^{\natural }(Y, K; \Gamma _{\sigma })$ as the reduced variant is justified by its relationship to (for example) the reduced version of Bar-Natan homology, as described in Corollary 6.6. We shall see later that, while $I^{\sharp }(U ; \Gamma _{\sigma })$ for the unknot $U$ is a free module of rank 2, the reduced version has rank $1$. As in [Reference Kronheimer and MrowkaKM12], there is an exact triangle relating the reduced homology (twice) to the unreduced homology.

2.6 The Künneth theorem for reduced homology

Given links $K_{1}$ and $K_{2}$, each with a framed base point, there is a natural construction of the connected sum $K_{1}\mathbin {\#} K_{2}$, also as a link with a framed base point. To spell this out, let $(\epsilon _{1}, \epsilon _{2}, \epsilon _{3})$ be the framing at the base point of $K_{1}$, with $\epsilon _{1}$ pointing along the knot. Using the framing, parametrize a standard ball $B_{3,1}$ around the base point. Construct $B_{3,2}$ similarly. Remove the interiors and form the connected sum by identifying the $2$-sphere $\partial B_{3,1}$ with $\partial B_{3,2}$ using the orientation-reversing diffeomorphism given by reflection in the $\epsilon _{1}$ direction. Take the base point on the new link to be the image of the point $\epsilon _{1}$ on $\partial B_{3,1}$.

The construction of $K_{1}\mathbin {\#} K_{2}$ from the two framed knots is functorial. That is, given cobordisms $S_{i}$ from $K_{i}'$ to $K_{i}$ for $i=1,2$, and given framed arcs $\gamma =(\gamma _{1}, \gamma _{2})$ joining the framed base points, we can form a cobordism

\[ S_{1} \mathbin{\#}_{\gamma} S_{2} \]

from $K'_{1}\mathbin {\#} K_{2}'$ to $K_{1}\mathbin {\#} K_{2}$ by performing the connected-sum construction in an interval family. The reduced instanton homology for a connected sum of framed knots is described as a tensor product by a Künneth theorem.

Proposition 2.4 Let $(C_{1}, \partial )$ and $(C_{2},\partial )$ be the differential $\mathcal {S}$-modules arising from the Floer complexes for the homology groups $I^{\natural }(K_{1};\Gamma _{\sigma })$ and $I^{\natural }(K_{2};\Gamma _{\sigma })$. Then the Floer complex for $K_{1}\mathbin {\#} K_{2}$ is chain-homotopy equivalent to the tensor product $C_{1}\otimes _{\mathcal {S}} C_{2}$. In particular, if $\mathcal {S}$ is a principal ideal domain, then there is a split exact sequence of $\mathcal {S}$-modules,

(24)\begin{align} &0 \longrightarrow I^{\natural}(K_{1};\Gamma_{\sigma}) \otimes_{\mathcal{S}} I^{\natural}(K_{2};\Gamma_{\sigma}) \longrightarrow I^{\natural}(K_{1}\mathbin{\#} K_{2};\Gamma_{\sigma})\nonumber\\ &\,\,\,\longrightarrow \mathrm{Tor}^{\mathcal{S}}_{1} (I^{\natural} (K_{1};\Gamma_{\sigma} ), I^{\natural}(K_{2};\Gamma_{\sigma}) ) \longrightarrow 0. \end{align}

The exact sequence, but not the splitting, is natural with respect to the maps induced by cobordisms $S_{1}$, $S_{2}$ and $S_{1}\mathbin {\#}_{\gamma }S_{2}$ as constructed above.

Proof. This a standard application of excision, as we now describe. The symmetries in the argument are more apparent in a more general version, so we consider four pairs $(Y_{i}, K_{i})$, $k=1,\ldots ,4$, where each $K_{i}$ is a based link. For each $i\ne j$, there is a connect sum of pairs,

\[ (Y_{ij},K_{ij}) = (Y_{i} , K_{i}) \mathbin{\#} (Y_{j}, K_{j}), \]

where the 3-manifolds and the links are both summed at the base points. Let $C^{\natural }_{ij}$ denote the chain group of free $\mathcal {S}$-modules arising as the instanton Floer complex for this connected sum of based pairs, with coefficients in the local system $\Gamma _{\sigma }$. The more general statement is then that there is a chain-homotopy equivalence,

(25)\begin{equation} C^{\natural}_{12} \otimes_{\mathcal{S}} C^{\natural}_{34} \simeq C^{\natural}_{13} \otimes_{\mathcal{S}} C^{\natural}_{24}, \end{equation}

and that the resulting maps on homology are natural for cobordisms. The statement of the original proposition arises as a special case, when each $Y_{i}$ is $S^{3}$, and $K_{3}$ and $K_{4}$ are both the unknot, so that $(Y_{13}, K_{13})=(Y_{1}, K_{1})$, $(Y_{24}, K_{24})=(Y_{2}, K_{2})$ and $C^{\natural }_{34}=\mathcal {S}$.

We now recall Floer's excision argument, particularly in the versions described in [Reference Kronheimer and MrowkaKM19b, Proposition 4.2] and [Reference Kronheimer and MrowkaKM19a, Proposition 3.3]. Let $U$ be as in Figure 2, a two-dimensional cobordism from the one-dimensional manifold-with-boundary $I\cup I$ to $I\cup I$. Take the product with $S^{2}$ to obtain a cobordism from $I\times S^{2}\cup I\times S^{2}$ to $I\times S^{2} \cup I\times S^{2}$. Then attach four copies of $[0,1]\times B^{3}$ to obtain a cobordism $W$ from $S^{3}\cup S^{3}$ to $S^{3}\cup S^{3}$. Inside $W$ there is an embedded foam, $\Phi$, formed from three copies of $U$. The pair $(W,\Phi )$ is a cobordism

\[ (S^{3},\theta) \cup (S^{3},\theta)\quad \text{to}\quad (S^{3},\theta) \cup (S^{3},\theta). \]

On one facet of the three facets of $\Phi$, let $\gamma _{i}$, $i=1,\ldots ,4$, be four arcs as shown in Figure 3. A regular neighborhood of $\gamma _{i}$ in $(W,\Phi )$ is a standard pair $[0,1]\times (B^{3}, B^{1})$ along which we form a sum with $[0,1]\times (Y_{i}, K_{i})$. The result is a cobordism of pairs, $(X,\Psi )$, from

\[ \big(Y_{12}, K_{12}^{\natural}\big) \cup \big(Y_{34}, K_{34}^{\natural}\big) \quad \text{to}\quad \big(Y_{13}, K_{13}^{\natural}\big) \cup \big(Y_{24}, K_{24}^{\natural}\big). \]

As in the proof of [Reference Kronheimer and MrowkaKM19a, Proposition 3.3], this cobordism of pairs gives rise to a map on the instanton chain complexes with local coefficients, in this case a chain map

\[ C^{\natural}_{12} \otimes_{\mathcal{S}} C^{\natural}_{34} \to C^{\natural}_{13} \otimes_{\mathcal{S}} C^{\natural}_{24}. \]

Figure 2.

The cobordism $U$ from two copies of an interval $I$ to another two.

Figure 3.

The foam $\Phi \subset W$ and the four arcs along which the pairs $[0,1]\times (Y_{i},K_{i})$ are summed.

By the same construction, a map in the other direction is constructed. The fact that the composite of the two, in either order, is chain-homotopic to the identity is proved by the usual argument, as in [Reference Kronheimer and MrowkaKM19a] for example.

3.1 Operators from characteristic classes of the base-point bundle

Given a point $y$ in the smooth part of the orbifold $(Y, K)$, there is a base-point $\mathop {{SO}}(3)$-bundle $\mathbb{E}_{y}$ on the configuration space, with Stiefel–Whitney classes

\[ \mathsf{w}_{1} , \mathsf{w}_{2}, \mathsf{w}_{3} \in H^{*}\big(\mathcal{B}^{\sharp}(K)_{\omega}; \mathbb{Z}/2\big). \]

(See [Reference Kronheimer and MrowkaKM19a, § 4].) For the instanton homology $I^{\sharp }(K ;\Gamma _{\sigma })$, these characteristic classes give rise to linear operators,

\[ w_{i} : I^{\sharp}(K ; \Gamma_{\sigma})_{\omega} \to I^{\sharp}(K ; \Gamma_{\sigma})_{\omega}. \]

The definition of these for the similar case of $J^{\sharp }(K ; \Gamma _{\theta })$ is presented in [Reference Kronheimer and MrowkaKM19a] and needs essentially no change. As in [Reference Kronheimer and MrowkaKM19a], we have the following result.

3.2 A two-dimensional cohomology class

Let $a$ be an arc in $Y$ with end points $\{p,q\}$ on $K\cup \theta$. The interesting case will be when $p$ and $q$ lie on different components, for example on $K$ and $\theta$, respectively. We require that $p$ and $q$ lie on the interior of edges, not at the vertices of the graph. We also require that $p$ and $q$ do not lie at end points of $\omega$ (if any). There is a universal $\mathbb{R}^{3}$ bundle

\[ \mathbb{E} \to a \times \mathcal{B}^{\sharp}(K)_{\omega}. \]

The restriction, $\mathbb{E}_{p}$, of $\mathbb{E}$ to the end point $\{p\} \times \mathcal {B}^{\sharp }(K)_{\omega }$ carries an involution on the $\mathbb{R}^{3}$ fibers, because of the $\mathbb{Z}/2$ stabilizer at this singular point of the orbifold, so $\mathbb{E}_{p}$ contains a distinguished real line subbundle, the $+1$ eigenspace of the involution:

\[ \mathbb{L}_{p} \subset \mathbb{E}_{p} \to \mathcal{B}^{\sharp}(K)_{\omega}. \]

This line bundle is trivial. Indeed, we have the following result.

With the lemma in mind, we introduce the following notation.

Suppose now that $p$ and $q$ are dots; let us return to the arc $a$ joining them as introduced above. The dot $p$ determines a distinguished section of $\mathbb{L}_{p}$ and hence a distinguished section $i_{p}$ of the $\mathbb{R}^{3}$ bundle $\mathbb{E}_{p}$. Similarly, using the dot $q$, we obtain a distinguished section $i_{q}$. Changing the sign of the second one, we obtain a distinguished section

\[ I = (i_{p} , -i_{q}) \]

of the restriction of $\mathbb{E}$ to $\partial a \times \mathcal {B}^{\sharp }(K)$.

The distinguished section on the boundary allows us to define a relative Euler class or a top Stiefel–Whitney class

\begin{align*} {\mathbb w}_{3} &= w_{3}(\mathbb{E} , I) \\ & \in H^{3} ((a, \partial a ) \times \mathcal{B}^{\sharp}(K) ; \mathbb{F}_{2} ). \end{align*}

We now take the slant product with the relative fundamental class of the arc to obtain a class on $\mathcal {B}^{\sharp }(K)_{\omega }$.

Definition 3.4 For an arc $a$ as above whose end points $p$ and $q$ are dots, we define a two-dimensional cohomology class with $\mathbb{F}_{2}$ coefficients on $\mathcal {B}^{\sharp }(K)_{\omega }$ as

(26)\begin{align} \lambda &= {\mathbb w}_{3} / [a, \partial a] \nonumber\\ &\in H^{2} ( \mathcal{B}^{\sharp}(K) ; \mathbb{F}_{2} ). \end{align}

The next lemma shows that the arc $a$ itself plays only an auxiliary role in this construction.

Proof. Consider the universal $\mathbb{R}^{3}$ bundle

\[ \mathbb{E} \to (Y{\setminus} K) \times \mathcal{B}^{\sharp}(K)_{\omega}. \]

The assertion to be proved is equivalent to the statement that

\begin{align*} w_{3}(\mathbb{E}) / [b] & = 0 \\ &\in H^{2} ( \mathcal{B}^{\sharp}(K)_{\omega} ; \mathbb{F}_{2} ) \end{align*}

for any 1-cycle $b$ in $Y{\setminus} K$. The bundle has trivial $w_{2}$ on $Y{\setminus} K$, so each irreducible connection lifts to an $\mathop {{SU}}\nolimits (2)$ connection with stabilizer $\pm 1$. This means that $w_{2}(\mathbb{E})$ can be represented by a $2$-cocycle which is pulled back from $\mathcal {B}^{\sharp }(K)_{\omega }$. The class $w_{3}$ is obtained by applying a Bockstein homomorphism to $w_{2}$. So, the class $w_{3}(\mathbb{E})$ is also pulled back from $\mathcal {B}^{\sharp }(K)_{\omega }$. It follows that $w_{3}(\mathbb{E}) / [b]$ is zero for all $1$-cycles $b$ in $Y{\setminus} K$ (and incidentally all $2$-cycles also).

The lemma allows us to write the class as a function of the two dots,

(27)\begin{equation} \lambda_{pq} \in H^{2}\big(\mathcal{B}^{\sharp}(K) ; \mathbb{F}_{2} \big). \end{equation}

The next lemma asks how $\lambda _{pq}$ changes if we change the orientation $o_{q}$ at one end point: that is, we replace $q$ by $\bar {q}$. We introduce the following notation: we write

\[ \lambda'_{pq} = \lambda_{p\bar{q}}. \]

Lemma 3.6 Let $p$ and $q$ be dots, and let $\lambda _{pq}$ and $\lambda '_{pq}$ be the resulting classes, as above. Then these classes satisfy the relations

(28)\begin{equation} \begin{aligned} \lambda_{pq} + \lambda'_{pq} & = w_{2}(\mathbb{E}_{q}), \\ \lambda_{pq} \lambda'_{pq} & = 0. \end{aligned} \end{equation}

Proof. Because of the independence of the choice of arc, and the way the signs are used in the definition of $I$ above, the first relation is equivalent to saying that

\[ \lambda_{qq} = w_{2}(\mathbb{E}_{q}), \]

where the left-hand side can be computed using the constant arc from $q$ to $q$.

The general statement at the level of characteristic classes is the following. Suppose that we have an $\mathbb{R}^{3}$ bundle $E\to T$ with a section $i_{0}$. Consider the pull-back $\pi ^{*}(E)$ to $[0,1]\times T$ with a section $I$ which is equal to $i_{0}$ on $\{0\}\times T$ and $-i_{0}$ on $\{1\}\times T$. Then the result of slanting $w_{3}(\pi ^{*}(E), I)$ with the fundamental class of $[0,1]$ is $w_{2}(E)$:

\[ w_{3}(\pi^{*}(E), I)/ [0,1] = w_{2}(E). \]

This can be verified by pulling back $E$ to $S^{1}\times T$ and tensoring by the Möbius bundle $\mu$ on $S^{1}$, in which case the assertion is

\[ w_{3}(\pi^{*}(E)\otimes \mu )/ [S^{1}] = w_{2}(E). \]

In this form, the verification is straightforward, using the splitting principle. This completes the proof of the first relation.

To set the second relation in a more general context, consider again an $\mathbb{R}^{3}$ bundle $E\to T$ with two non-vanishing sections $i_{0}$ and $i_{1}$. Let $I$ be a path of sections, from $i_{0}$ to $-i_{1}$, through sections which may vanish: we take explicitly

\[ I(t) = (1-t) i_{0} - t i_{1}. \]

Similarly, let $I'(s)$ be the path from $i_{0}$ to $i_{1}$ given by

\[ I'(s) = (1-s) i_{0} + s i_{1}. \]

We have cohomology classes by

\begin{align*} \lambda&=w_{3}(E,I)/[0,1]\\ & \in H^{2}(T;\mathbb{F}_{2}), \\ \lambda'&= w_{3}(E, I')/[0,1]\\ & \in H^{2}(T;\mathbb{F}_{2}), \end{align*}

where we now interpret $I$ and $I'$ as sections on $[0,1]\times T$ that are non-zero at the boundaries. To show that $\lambda \lambda '=0$, it is sufficient to show that there is no $(t,s)$ in the interior of $[0,1]\times [0,1]$ for which the sections $I(t)$ and $I'(s)$ have a common zero in $T$. A necessary condition for a common zero is that the determinant of the matrix

\[ \begin{pmatrix} (1-t) & -t \\ (1-s) & s \end{pmatrix} \]

is zero. But the determinant is $s+t-2st$, which is strictly positive on the interior of $[0,1]\times [0,1]$. The result follows.

Corollary 3.7 The class $\lambda _{pq}$ satisfies the relation

\[ \lambda^{2}_{pq} + w_{2}(\mathbb{E}_{q}) \lambda_{pq} = 0. \]

3.3 Operators from the two-dimensional classes

In the usual way, and following the exposition in [Reference Kronheimer and MrowkaKM19a], the cohomology class $\lambda _{pq}$ gives rise to an operator

(29)\begin{equation} \Lambda_{pq} : I^{\sharp}(Y, K ; \Gamma_{\sigma})_{\omega} \to I^{\sharp}(Y, K ; \Gamma_{\sigma})_{\omega}. \end{equation}

In a little more detail, let $a$ be the chosen arc joining the two dots, regarded as subset of the cylinder $\check X = \mathbb{R}\times \check Y$, in the slice where the $\mathbb{R}$ coordinate is zero. Following [Reference Kronheimer and MrowkaKM19a, § 4.3], let $Z\subset \check X$ be a subset of $\check X$ which includes a neighborhood of $a$ and such that the restriction map

\[ H^{1}(Y{\setminus} K; \mathbb{F}_{2}) \to H^{1}(Z{\setminus} K; \mathbb{F}_{2}) \]

is injective. The latter condition means that there is a well-defined restriction map for marked connections,

\[ M(\alpha, \beta) \to \mathcal{B}^{*}(Z ; \mu_{Z}), \]

where $\mu _{Z}$ is the intersection with $Z$ of the marking data $\mathbb{R}\times \mu _{\omega }$. Because $Z$ contains a neighborhood of $a$, the class $\lambda _{pq}$ can be defined on $\mathcal {B}^{*}(Z ; \mu _{Z})$, where it is dual to a stratified codimension-2 subvariety $V$. The matrix entries of $\Lambda _{pq}$ at the chain level are defined by counting points of the intersections

\[ M(\alpha, \beta) \cap V \]

and weighting them using the local system. As in [Reference Kronheimer and MrowkaKM19a], the necessary compactness results hold because the cohomology class has dimension $2$ and $Z$ can be chosen so that it meets the foam only in the faces (at neighborhoods of $p$ and $q$).

Using notation that is parallel to the notation for $\lambda _{pq}$ and $\lambda '_{pq}$, we write $\Lambda '_{pq}$ for the operator of the same form as (29), but using $\bar {q}$ in place of $q$. The relations in Lemma 3.6, satisfied by $\lambda _{pq}$ and $\lambda '_{pq}$, give rise to relations satisfied by the corresponding operators on Floer homology.

Lemma 3.8 Let $\Lambda _{pq}$ and $\Lambda '_{pq}$ be the operators on $I^{\sharp }(K;\Gamma _{\sigma })_{\omega }$ arising from a pair of dots $\{p,q\}$ as above. Then these operators satisfy the relations

(30)\begin{equation} \begin{aligned} \Lambda_{pq} + \Lambda'_{pq} & = \sigma(P), \\ \Lambda_{pq} \Lambda'_{pq} & = \sigma(Q_{pq}), \end{aligned} \end{equation}

where $P$ is the element of $\mathcal {R}$ given by (5), and

\[ Q_{pq} = \big(T_{m_{p}}^{2} + T_{m_{p}}^{{-}2}\big) + \big(T_{m_{q}}^{2} + T_{m_{q}}^{{-}2}\big). \]

In the above formula, $T_{m_{p}}$ and $T_{m_{q}}$ are the variables from $\{T_{0}, T_{1}, T_{2}, T_{3}\}$ associated to the edges of $K \cup \theta$ on which $p$ and $q$ lie. Thus, $m_{p}=1, 2$ or $3$ if $p$ lies on the edge $e_{1}$, $e_2$ or $e_{3}$ of $\theta$, and $m_{p}=0$ if $p$ lies on $K$.

Proof. By an excision argument [Reference Kronheimer and MrowkaKM12], it is sufficient to prove this in the case that $\omega$ is empty. It is then sufficient to consider the case that the base change $\sigma$ is the identity.

The first of the two relations follows from the corresponding formula for $\lambda _{pq} + \lambda '_{pq}$ in Lemma 3.6, together with the formula $w_{2}=P$ from Lemma 3.1.

The second relation, for the product, is more subtle, because it involves a four-dimensional moduli space, and there is a contribution from codimension-4 bubbling which may occur at the end points $p$ and $q$ of the arc $a \subset Z$.

As in [Reference XieXie16] and [Reference Kronheimer and MrowkaKM19a], the contributions from the bubbles at $p$ and $q$ are universal quantities, so that the relation for the product has the general shape

\[ \Lambda_{pq} \Lambda'_{pq} = F(T_{m_{p}}) + F(T_{m_{q}}), \]

where $F$ is universal and is a finite Laurent series in one variable. To compute $F$, we take as a special case the situation that $K$ is empty and $p$ and $q$ lie on the edges $e_{2}$ and $e_{1}$ of $\theta$, respectively. In the ring $\mathbb{F}_{2}[T_{1}^{\pm 1}, T_{2}^{\pm 1}, T_{3}^{\pm 1}]$, we then have elements $\Lambda$ and $\Lambda '$ with relations

\begin{align*} \Lambda + \Lambda' &= P \\ &= T_{1}T_{2}T_{3} + T_{1}T_{2}^{{-}1}T_{3}^{{-}1} + T_{2}T_{3}^{{-}1}T_{1}^{{-}1} + T_{3}T_{1}^{{-}1}T_{2}^{{-}1} \end{align*}

and

\[ \Lambda \Lambda' = F(T_{2}) + F(T_{1}). \]

The only way to solve the constraint that $\Lambda \Lambda '$ is a function of $T_{1}$ and $T_{2}$ only is to have the general shape

\begin{align*} \Lambda & = T_{3}^{a} G(T_{1}, T_{2}), \\ \Lambda'& = T_{3}^{{-}a} H(T_{1}, T_{2}) \end{align*}

for some Laurent polynomials $G$ and $H$ in two variables. The shape of the formula for $\Lambda +\Lambda '$ tells us that $a$ must be $\pm 1$ and that $\Lambda$ and $\Lambda '$ must consist of the corresponding monomials from the formula for $P$. Thus,

(31)\begin{equation} \begin{aligned} \Lambda & = T_{3}\big(T_{1}T_{2} + T_{1}^{{-}1}T_{2}^{{-}1}\big), \\ \Lambda' & = T_{3}^{{-}1}\big(T_{1}^{{-}1}T_{2} + T_{1}T_{2}^{{-}1}\big) \end{aligned} \end{equation}

or vice versa. Either way, we have $F(T)= T^{2} + T^{-2}$.

As with the cohomology classes themselves, we have an immediate corollary of the lemma, for the operator $\Lambda _{pq}$.

Corollary 3.9 The operator $\Lambda _{pq}$ satisfies the relation

\[ \Lambda^{2}_{pq} +\sigma( P) \Lambda_{pq} + \sigma(Q_{pq}) = 0. \]

To create an operator that treats the three edges of $\theta$ symmetrically, we make the following definition.

Definition 3.10 Fix once and for all a dot $p_{m}$ on each edge $e_{m}$ of $\theta$ for $m=1,2,3$. Then, given a dot $q$ on the link $K$, we define

\[ \Lambda_{q} = \Lambda_{p_{1}q} + \Lambda_{p_{2}q} + \Lambda_{p_{3}q}, \]

with $\Lambda '_{q}$ defined similarly.

Corollary 3.11 The operator $\Lambda _{q}$ above satisfies the relation

\[ \Lambda^{2}_{q} + \sigma(P) \Lambda_{q} +\sigma( Q) = 0, \]

where $P$ and $Q$ are given by the formulae in (5). Furthermore, $\Lambda _{q} + \Lambda '_{q}= \sigma (P)$.

3.4 Surfaces with dots

The operators $\Lambda _{q}$ on $I^{\sharp }(K;\Gamma _{\sigma })_{\omega }$ that we have defined can be combined, in the usual way, with the functorial maps obtained from cobordisms $S$ between knots and links. Thus, suppose that we are given a cobordism $(X,S)$ from $(Y_{0},K_{0})$ to $(Y_{1}, K_{1})$ as in (23), and let $q$ be a dot on $S$. As before, this means a point with a choice of orientation $o_{q}$ of the line bundle $\mathbb{L}_{q}$. We then obtain a map

(32)\begin{equation} I^{\sharp}((S;q) ; \Gamma_{\sigma})_{\omega} : I^{\sharp}( K_{0}; \Gamma_{\sigma})_{\omega_{0}} \to I^{\sharp}( K_{1}; \Gamma_{\sigma})_{\omega_{1}}. \end{equation}

If $q$ can be joined by a path on $S$ to a point $q_{0}\in K_{0}$ (respectively, a point $q_{1}\in K_{1}$), then this map is equal to the composite,

\[ I^{\sharp}(S ; \Gamma_{\sigma})_{\omega} \circ \Lambda_{q_{0}}, \]

respectively,

\[ \Lambda_{q_{1}} \circ I^{\sharp}(S ; \Gamma_{\sigma})_{\omega}. \]

The functorial properties of $I^{\sharp }$ extend to this larger category in which the morphisms are ‘cobordisms with dots’. We note that, as with the case of dots on a link $K$, an orientation of the line bundle $\mathbb{L}_{q}$ is equivalent to a choice of orientation for a neighborhood of $q$ in $S$. So, a dot can be regarded as point in $S$ together with an orientation of $T_{q}S$.

4.1 Connect sum with $\mathbb{R}\mathbb{P}^{2}$

In $S^{4}$, there are two standard copies of $\mathop {\mathbb{R}\mathbb{P}}\nolimits ^{2}$ (see [Reference Kronheimer and MrowkaKM12], for example), which we call $R_{+}$ and $R_{-}$. These have self-intersection numbers

\begin{align*} R_{+} \cdot R_{+} &={+}2, \\ R_{-} \cdot R_{-} &= -2. \end{align*}

From $(X,S)$, we can form a new cobordism as a connected sum,

\[ (X,\tilde{S})= (X,S) \mathbin{\#} (S^{4}, R_{{\pm} }). \]

Lemma 4.1 In the case $\tilde {S} = S\mathbin {\#} R_{+}$, or $\tilde {S} = S \mathbin {\#} R_{-}$, we have

\[ I^{\sharp}(X,\tilde{S} ; \Gamma_{\sigma})_{\omega} = 0. \]

We postpone the proof until after the statement of the next lemma.

If $\sigma (T_{0})=1$, then we can use more general representatives for classes $w_{2}$. In particular, a circle representing the generator of $H_{1}(R_{\pm })$ bounds a disk in the complement of $R_{\pm }$ in $S^{4}$. Let us write $\pi$ for this disk. It represents a non-zero mod-2 class in the homology of the complement. In the complement of $\tilde {S} = S \mathbin {\#} R_{\pm }$, we can then use the Stiefel–Whitney class represented by $\omega +\pi$.

Lemma 4.2 Suppose that $\sigma (T_{0})=1$ in the ring $\mathcal {S}$. Then, in the case that $\tilde {S} = S\mathbin {\#} R_{+}$, we have

\[ I^{\sharp}(X,\tilde{S} ; \Gamma_{\sigma})_{\omega+\pi} = I^{\sharp}(X,S;\Gamma_{\sigma})_{\omega}. \]

In the case that $\tilde {S} = S\mathbin {\#} R_{-}$, we have

\[ I^{\sharp}(X,\tilde{S} ; \Gamma_{\sigma})_{\omega+\pi} = \sigma(P)\, I^{\sharp}(X,S;\Gamma_{\sigma})_{\omega}, \]

where $P \in \mathcal {R}$ is the element given by the formula (5).

Proofs of the two lemmas. Proofs of the two lemmas

There are four assertions altogether: two surfaces $R_{\pm }$ and two choices of Stiefel–Whitney class. In each case, we have a connected sum with $(S^{4}, R_{\pm })$ along $(S^{3}, S^{1})$. We apply the usual stretching argument and we consider the possible weak limit on $(S^{4}, R_{\pm })$. The gluing parameter in the connected sum is $S^{1}$, so we will have non-zero contributions only when the weak limit on $(S^{4}, R_{\pm })$ is an anti-self-dual connection with $S^{1}$ stabilizer. However, there are no non-zero harmonic 2-forms on these orbifolds, so the only possibility is a flat connection. There is a unique flat $SO(3)$ bifold connection $[A_{\pm }]$ on $(S^{4}, R_{\pm })$ because the fundamental group of the complement is cyclic of order $2$. Its Stiefel–Whitney class is represented by $\pi$. This proves the first lemma: there is no contribution for the Stiefel–Whitney class $\omega$.

The anti-self-dual connection $[A_{\pm }]$ is unobstructed in the case of $R_{+}$ and has a two-dimensional obstruction space in the case of $R_{-}$, as explained in [Reference Kronheimer and MrowkaKM12, § 2.7]. So, for $R_{+}$, we have

\[ I^{\sharp}(X,S \mathbin{\#} R_{+} ; \Gamma_{\sigma})_{\omega+\pi} = I^{\sharp}(X,S;\Gamma_{\sigma})_{\omega}. \]

In the case of $R_{-}$, we can identify the two-dimensional gluing obstruction with the 2-plane bundle $\eta$, and the effect of gluing is the same as cutting down by $w_{2}(\eta )$. Lemma 3.1 tells us that this is multiplication by $\sigma (P)$.

4.2 Connect sum with $T^{2}$

Let $T$ be a standard unknotted torus in $\mathbb{R}^{3}$ and regard $T$ by inclusion as a submanifold of $S^{4}$. We may form a connected sum

\[ \big(X, \tilde{S}\big) = (X,S) \mathbin{\#} \big(S^{4}, T\big). \]

Lemma 4.3 When $(X,\tilde {S})$ is formed from $(X,S)$ by a connected sum with the standard torus $T$ as above, we have

\[ I^{\sharp}\big(X,\tilde{S} ; \Gamma_{\sigma}\big)_{\omega} = \sigma( P)\, I^{\sharp}(X,S;\Gamma_{\sigma})_{\omega}. \]

4.3 Double points and blowing up

As in [Reference Kronheimer and MrowkaKM11] and [Reference Kronheimer and MrowkaKM13], we can extend the definition of the maps $I^{\sharp }(X,S;\Gamma _{\sigma })_{\omega }$ induced by cobordisms to include also the case that $S$ is a normally immersed surface in $X$. Our approach in the present paper is a slight variant of what was done in the two previous cited papers: what we will do here is better adapted to the case of an unoriented surface $S$.

So, let $f : S \looparrowright X$ be an ‘immersed cobordism’ from $(Y_{0}, K_{0})$ to $(Y_{1}, K_{1})$. We always assume, as in [Reference Kronheimer and MrowkaKM11], that $f$ has only transverse double points, and that these are in the interior of $X$. That is, the surface is normally immersed. We also assume that the double points do not lie on the surface $\omega$ which represents $w_{2}$. We do not want to orient $S$, and we therefore do not give a sign ${\pm }1$ to the double points of the immersion. At a double point $x\in f(S)$, we may choose the metric on $X$ so that the two branches of the immersion have orthogonal tangent planes, $\pi$ and $\pi '$, in $T_{x}X$. There are then exactly two complex structures $J$ and $-J$ on $T_{x}X$ such that:

1. (i) the complex structure is compatible with the metric and orientation of $T_{x}X$;

2. (ii) $\pi$ and $\pi '$ are $J$-invariant.

The blow-ups of $X$ at $x$ with respect to the complex structures $J$ and $-J$ are canonically identified: in both blow-ups, the exceptional set $\epsilon \subset \tilde {X}$ is the set of $J$-invariant 2-planes in $T_{x}X$. When identified with $\mathbb{C}\mathbb{P}^{1}$, however, the complex orientation of the exceptional set is different in the two cases. The proper transform $\tilde {f} : \tilde {S} \looparrowright \tilde {X}$ has one fewer double point than $f$.

In the above situation, we define $I^{\sharp }(X, S ; \Gamma )_{\omega }$ for the immersed cobordism by requiring that

(34)\begin{equation} I^{\sharp}\big(X, S ; \Gamma_{\sigma}\big)_{\omega} = I^{\sharp}\big(\tilde{X}, \tilde{S} ; \Gamma_{\sigma}\big)_{\omega} + I^{\sharp}\big(\tilde{X}, \tilde{S} ; \Gamma_{\sigma}\big)_{\omega+\epsilon}. \end{equation}

On the right, we see the proper transform, equipped with two different Stiefel–Whitney classes, differing by the exceptional set $\epsilon$ of the blow-up. By applying the definition to each double point in turn, we arrive at a definition that reduces to the standard case of embedded cobordisms.

Before proceeding further, we make some remarks about this definition. The proper transform is being used here to construct a functor from a category in which the morphisms are immersed cobordisms to one in which the morphisms are embedded cobordisms. In the previous papers [Reference Kronheimer and MrowkaKM11, Reference Kronheimer and MrowkaKM13, Reference KronheimerKro97], such a construction was used with only the first of the two terms on the right. The reason for using the two terms, involving both $\omega$ and $\omega +\epsilon$, is to provide a deformation invariance that would otherwise be absent, in the case that $\omega$ has boundary on $S$. To understand this, consider a local model for a double point of $S$, consisting of a pair of disks $D_{1}\cup D_{2}$ in the product $D_{1}\times D_{2}$, and let coordinates $(x_{i}, y_{i})$ be standard coordinates on $D_{i}$, so that the disks meet at the origin. Let $\omega$ be described in this neighborhood by

\[ \omega = \{y_{1} = 1/2, \; y_{2} = 0, \; x_{2}\ge 0\}, \]

so that $\partial \omega$ is the line $y_{1}=1/2$ on the disk $D_{1}$. Let $\omega '$ be obtained from $\omega$ by deforming $\omega$ in this neighborhood in such a way that (i) $\partial \omega '$ is the line $y=-1/2$ on $D_{1}$; and (ii) $\omega '$ intersects $D_{2}$ transversely at a point. Let $(\tilde {X}, \tilde {S}$ be obtained by blowing up the double point $D_{1}\cap D_{2}$, and regard $\omega$,$\omega '$ as lying in $\tilde {X}$. In this situation (assuming that this local picture is just part of cobordism of pairs), we have

\[ I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega} \ne I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega'} \]

in general, because $\omega$ and $\omega '$ are representatives of Stiefel–Whitney classes of different orbifold bundles. Instead, we have

\[ I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega} = I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega'+\epsilon} \]

and similarly

\[ I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega+\epsilon} = I^{\sharp}(\tilde{X}, \tilde{S} ; \Gamma_{\sigma})_{\omega'}. \]

So, if we wish to define $I^{\sharp }( X, S ; \Gamma _{\sigma })_{\omega }$ when $S$ is normally immersed, and if we wish the result to be independent of the choice of $\omega$, in this way, we should take the two terms together, as we have done in (34).

With that said, if we impose the restriction that we consider only $\omega$ without boundary along $S$, then we are free to modify the definition of the functor: for any fixed choice of $\xi \in \mathcal {S}$, we can define a functor $I^{\sharp }_{\xi }$ by leaving everything unchanged except for the rule for dealing with double points, where we substitute the variant

(35)\begin{equation} I^{\sharp}_{\xi}\big(X, S ; \Gamma_{\sigma}\big)_{\omega} = I^{\sharp}_{\xi}\big(\tilde{X}, \tilde{S} ; \Gamma_{\sigma}\big)_{\omega} + \xi \,I^{\sharp}_{\xi}\big(\tilde{X}, \tilde{S} ; \Gamma_{\sigma}\big)_{\omega+\epsilon}. \end{equation}

For the rest of this paper, we shall remain with the more restricted case $\xi =1$, with only occasional comments about the more general version.

4.4 Twist moves and finger moves

We now follow the strategy from [Reference Kronheimer and MrowkaKM11] to see how $I^{\sharp }(X, S ; \Gamma _{\sigma })$ changes when the immersion is changed in three standard ways (introducing additional double points). These are the ‘twist move’, which comes in two oriented flavors, and the ‘finger move’.

Proposition 4.4 (cf. [Reference Kronheimer and MrowkaKM11, Proposition 5.2] and [Reference Kronheimer and MrowkaKM13, Proposition 3.1])

Let $S^{*}$ be obtained from $S$ by either a positive twist move or a finger move (introducing a canceling pair of double points). Then we have

\[ I^{\sharp}(X, S^{*} ; \Gamma_{\sigma})_{\omega} = \sigma(L)\, I^{\sharp}(X, S ; \Gamma_{\sigma})_{\omega}, \]

where

\[ L = P + T_{0}^{2} + T_{0}^{{-}2}. \]

For the negative twist move, on the other hand, the map $I^{\sharp }$ is unchanged:

\[ I^{\sharp}(X, S^{*} ; \Gamma_{\sigma})_{\omega} = I^{\sharp}(X, S ; \Gamma_{\sigma})_{\omega}. \]

We prove the various parts of this proposition in the paragraphs. Our exposition describes just the case of $\Gamma$, because the results are local, and the general $\Gamma _{\sigma }$ is obtained by base change.

4.4.1 Twist moves

We begin with the positive twist move. In this case, as explained in [Reference Kronheimer and MrowkaKM11] and [Reference KronheimerKro97], the result of the positive twist move followed by taking the proper transform in the blow-up is to replace $(X,S)$ with

\[ (X' , S') = (X, S) \mathbin{\#} (\overline{\mathbb{C}\mathbb{P}}^{2} , C), \]

where $C$ is a smooth conic curve. According to our definition (34), we must therefore compute

\[ I^{\sharp}(X', S' ; \Gamma)_{\omega} + I^{\sharp}(X', S' ; \Gamma)_{\omega+\epsilon}. \]

Once again, we compute by a connected-sum argument. A dimension count shows that the weak limit $[E,A]$ on $(\overline {\mathbb{C}\mathbb{P}}^{2} , C)$ lies in a moduli space of formal dimension $d_{0} \le -1$, which means that its action $\kappa _{0}$ satisfies the bound $\kappa _{0} \le 1/4$. Since the formal dimension is negative, the connection must be reducible, to $\pm 1$, to $\mathop {{SO}}(2)$ or to $O(2)$. The double cover is $S^{2}\times S^{2}$, with the involution $\tau (x,y) = (-y,-x)$. The fixed-point set is the anti-diagonal $\Delta ^{-}$. The pull-back $[\tilde {E}, \tilde {A}]$ on $S^{2}\times S^{2}$ must be reducible, either to $\mathop {{SO}}(2)$ or to the trivial group, so this $\mathop {{SO}}(3)$ bundle has the form

\[ \mathbb{R} \oplus K, \]

where $e(K)$ can be taken to be $\tau$-invariant in the case that $[E,A]$ reduces to ${\pm }1$, and $\tau$-anti-invariant in the case that $[E,A]$ reduces to $O(2)$. In the standard basis, $e(K)$ has the form $(\delta , -\delta )$ or $(\delta , \delta )$, respectively. A class of the second sort is not represented by an anti-self-dual form, however. So, $e(K)=(\delta , -\delta )$ and $[E,A]$ reduces either to $\pm 1$ or to $\mathop {{SO}}(2)$. The bound on $\kappa _{0}$ means that $e(K)^{2} \ge -2$, so $\delta ^{2}\le 1$.

If $\delta =0$, then $w_{2}(\tilde {E}) = 0$, which means that $w_{2}(E) = \epsilon$ in the neighborhood of the blow-up. If $\delta =\pm 1$, then $w_{2}(E)$ is zero in the neighborhood. The dimension count shows that the two cases $\delta ={\pm }1$ are unobstructed, and these contribute the terms $T_{0}^{2} + T_{0}^{-2}$. (The calculation here is just as in [Reference Kronheimer and MrowkaKM11].) So, we have

\[ I^{\sharp}(X', S' ; \Gamma)_{\omega} = (T_{0}^{2} + T_{0}^{{-}2}) I^{\sharp}(X, S ; \Gamma)_{\omega}. \]

The case $\delta =0$ is the case of the flat bifold connection on $(\overline {\mathbb{C}\mathbb{P}}^{2}, C)$ and it contributes to the term $I^{\sharp }( X', S' ; \Gamma )_{\omega +\epsilon }$. The obstruction space is again two dimensional, and just as the case of a connected sum with either $(S^{4}, R_{-})$ or $(S^{4}, T^{2})$, we obtain

\[ I^{\sharp}(X', S' ; \Gamma)_{\omega+\epsilon} = P I^{\sharp}(X, S ; \Gamma)_{\omega}. \]

This concludes the proof for the positive twist move.

The negative twist move is straightforward. In this case we must consider $(X', S')$ obtained from $(X,S)$ by forming the connect sum with $(\overline {\mathbb{C}\mathbb{P}}^{2}, \emptyset )$. The term with $\epsilon$ does not contribute, and the term $I^{\sharp }(X', S' ; \Gamma )_{\omega }$ is equal to $I^{\sharp }(X, S ; \Gamma )_{\omega }$ as in [Reference Kronheimer and MrowkaKM11].

4.4.2 Finger moves

Let $S^{*}$ be obtained from $S$ by a finger move, and let $S'$ be obtained from $S^{*}$ by blowing up at the two double points and taking the proper transform. Let $\epsilon _{1}$ and $\epsilon _{2}$ be the exceptional sets of the two blow-ups. From the definition in (34), we see that what we must compute is a sum of four terms

(36)\begin{equation} \mathop{{I}}\nolimits^{\sharp}(X', S' ; \Gamma)_{\omega} + \mathop{{I}}\nolimits^{\sharp}(X', S' ; \Gamma)_{\omega + \epsilon_{1}} + \mathop{{I}}\nolimits^{\sharp}(X', S' ; \Gamma)_{\omega + \epsilon_{2}} + \mathop{{I}}\nolimits^{\sharp}(X', S' ; \Gamma)_{\omega + \epsilon_{1} + \epsilon_{2}} \end{equation}

and the desired answer is $L \,I^{\sharp }(X, S; \Gamma )_{\omega }$, where $L$ is as in the proposition.

To focus on the region where the change occurs, let us write

\[ (X,S) = (X_{1}, S_{1}) \cup (X_{2}, S_{2}), \]

where $(X_{2}, S_{2})$ is a standard 4-ball containing a standard pair of disks, and $(X_{1}, S_{1})$ is the closure of the complement. The two pairs meet along a pair $(S^{3}, U_{2})$, where $U_{2}\subset$ is a standard two-component unlink. Let $(X_{2}', S_{2}')$ be obtained from $(X_{2}, S_{2})$ by the finger move and proper transform. So, we have

\[ (X', S') = (X_{1}, S_{1}) \cup (X_{2}' , S_{2}'). \]

The manifold $(X_{2}' , S_{2}')$ has boundary $(S^{3}, U_{2})$, and we can form from it a closed pair by attaching a 4-ball and a standard pair of disks. We write $(Z, \Sigma )$ for the resulting pair:

(37)\begin{equation} (Z, \Sigma) = \big(X_{2}', S_{2}'\big) \cup \big(B^{4}, D^{2}\amalg D^{2}\big). \end{equation}

The manifold $Z$ is a connected sum of two copies of $\overline {\mathbb{C}\mathbb{P}}^{2}$, and we write $E_{1}$, $E_{2}$ for the two exceptional curves. The surface $\Sigma$ is a union of two spheres,

\[ \Sigma = \Sigma_{1} \amalg \Sigma_{2}, \]

each of which has square $-2$. The two components $\Sigma _{1}$ and $\Sigma _{2}$ have the same mod-2 homology class, but over the integers we have (depending on choices made)

\begin{align*} [\Sigma_{1}] &={-} [E_{1}] - [E_{2}],\\ [\Sigma_{2}] &={-} [E_{1}] + [E_{2}]. \end{align*}

The proof of the formula for the finger move depends on understanding the moduli spaces on $(Z,\Sigma )$ for small energy $\kappa$, namely $\kappa =0$ and $\kappa =1/4$. As in (36) above, we will need to understand these moduli spaces

\[ M(Z,\Sigma)_{\nu} \]

for four different values of the Stiefel–Whitney class $\nu$, namely

\[ \nu = 0, \quad \nu =\epsilon_{1}, \quad \nu =\epsilon_{2} \quad \text{and}\quad \nu =\epsilon_{1} + \epsilon_{2}, \]

where $\epsilon _{i}$ is a representative in the class of $E_{i}$. Let us write

\[ M(Z, \Sigma)_{*} \]

for the union of $M(Z,\Sigma )_{\nu }$ over these four values of $\nu$.

Because we are working with $I^{\sharp }$, our interpretation of $M(Z,\Sigma )_{*}$ is that it parametrizes $\mathop {{SU}}\nolimits (2)$ gauge equivalence classes of anti-self-dual $\mathop {{SU}}\nolimits (2)$ connections on the complement of the four spheres $E_{1}$, $E_{2}$, $\Sigma _{1}$ and $\Sigma _{2}$, such that the limiting holonomy around the links of the spheres $\Sigma _{i}$ is order 4, and the holonomy around the links of the spheres $E_{i}$ are each $1$ or $-1$, depending on the value of $\nu$. The metric on $Z$ is an orbifold metric as usual, with singular set $\Sigma$.

Consider a flat line bundle $\xi$ on

\[ Z' = Z{\setminus} \big( \Sigma_{1}, \Sigma_{2}, E_{1}, E_{2}\big). \]

Write $(\sigma _{1}, \sigma _{2}, \eta _{1}, \eta _{2})$ for the holonomy of $\xi$ around the links of these four spheres, and require that these are $({\pm }1, {\pm }1, {\pm }1, {\pm }1)$. A push-off of $E_{1}$ meets $\Sigma _{1}$, $\Sigma _{2}$ and $E_{1}$ once each. Similarly with $E_{2}$. So,we have relations

\[ \eta_{1}= \eta_{2} = \sigma_{1} + \sigma_{2}. \]

So, there are four possibilities for $\xi$, including the trivial bundle, and their possible holonomies are

\begin{align*} (\sigma_{1}, \sigma_{2}, \eta_{1}, \eta_{2}) & = (1,1,1,1), \quad\text{or} \\ & = (1,-1,-1,-1), \quad\text{or} \\ & = ({-}1,1,-1,-1), \quad\text{or} \\ & = ({-}1,-1,1,1). \end{align*}

They form the group isomorphic to $V_{4}$.

The flat line bundles act $\xi$ act on $M(Z,\Sigma )_{*}$ by tensor product. So, we have an action of $V_{4}$ on this moduli space. Tensoring by $\xi$ either leaves $\nu$ unchanged (if $\sigma _{1}=\sigma _{2}$) or adds $\epsilon _{1} + \epsilon _{2}$. So, the subset

(38)\begin{equation} M(Z,\Sigma)_{0} \cup M(Z,\Sigma)_{\epsilon_{1} + \epsilon_{2}} \end{equation}

is closed under the Klein 4-group action, as is the complementary subset,

(39)\begin{equation} M(Z,\Sigma)_{\epsilon_{1}} \cup M(Z,\Sigma)_{\epsilon_{2}}. \end{equation}

The quotient

\[ M(Z,\Sigma)_{*} / V_{4} \]

parametrizes $\mathop {{SO}}(3)$ anti-self-dual connections $[B]$ on the complement of the four spheres with the property that the holonomy around the links of the $\Sigma _{i}$ has order $2$ and the holonomy around the links of the $E_{i}$ is $1$ in $\mathop {{SO}}(3)$. This is the same as the space of bifold $\mathop {{SO}}(3)$ connections (without marking) on $(Z,\Sigma )$:

\[ M(Z,\Sigma)_{*} / V_{4} = M_{\mathop{{SO}}(3)}(Z,\Sigma). \]

The next lemma (and the notation ‘twisted reducibles’) is from [Reference Kronheimer and MrowkaKM95].

Lemma 4.5 The action of the Klein 4-group is free except at ‘twisted reducibles’. That is, the $\mathop {{SU}}\nolimits (2)$ connection $A$ is gauge equivalent to $A\otimes \xi$ if and only if the holonomy of $[\mathrm {ad}(A)]$ is contained in $O(2)\subset \mathop {{SO}}(3)$ and the associated real line bundle to the $O(2)$ connection is isomorphic to $\xi$.

The situation described in the lemma above can happen only if $\xi$ either trivial or has

\[ \big(\sigma_{1}, \sigma_{2}, \eta_{1}, \eta_{2}\big) =({-}1,-1,1,1). \]

We are now ready to describe the small-action moduli spaces, beginning with a description of the quotients $M_{\mathop {{SO}}(3)}(Z,\Sigma )_{*}$.

Lemma 4.6 The space of bifold connections $M_{\mathop {{SO}}(3)}(Z,\Sigma )$ with $\kappa =0$ consists of a single point, with $\mathbb{Z}/2$ monodromy and $O(2)$ stabilizer.

For generic metrics, the space of bifold connections $M_{\mathop {{SO}}(3)}(Z,\Sigma )$ with $\kappa =1/4$ consists of a single arc and possibly some additional circles. Except for the end points of the arc, these bifold connections with $\kappa =1/4$ are irreducible (i.e. have trivial stabilizer in $\mathop {{SO}}(3)$). The end points of the arc have $\mathop {{SO}}(2)$ holonomy and therefore $\mathop {{SO}}(2)$ stabilizer.

Proof. For $\kappa =0$, we are looking at flat orbifold bundles, or $\mathop {{SO}}(3)$ representations of the orbifold fundamental group. The fundamental group of $Z{\setminus} \Sigma$ is $\mathbb{Z}/2$, because this space is $(0,1)\times \mathop {\mathbb{R}\mathbb{P}}\nolimits ^{3}$. The two links of $S$ are non-zero elements. So, there is a unique bifold connection.

For $\kappa =1/4$, consider the branched double cover of $\tilde {Z} \to Z$ along $\Sigma$. Let $\tilde {\Sigma }$ be the inverse image of $\Sigma$. This consists of two spheres $\tilde {\Sigma }_{i}$, each of self-intersection $-1$. The manifold $\tilde {Z}$ itself is diffeomorphic to a connected sum of two copies of $\overline {\mathbb{C}\mathbb{P}}^{2}$. On $\tilde {Z}$, we seek $\mathop {{SO}}(3)$ connections with action $\tilde \kappa =1/2$, which requires $w_{2}^{2} = 2$ mod $4$. So, $w_{2}=(1,1)$ in the standard basis. This is the sort of one-dimensional moduli space considered in [Reference Fintushel and SternFS84], from which we learn that the moduli space on $\tilde {Z}$ with $w_{2}=(1,1)$ and $\kappa =1/2$ is one dimensional and compact. Its end points correspond to pairs of integer classes $\pm \lambda$ where $\lambda ^{2}=-2$ and $\lambda =w_{2}$ mod $2$. The only possibilities are $\pm (1,1)$ and $\pm (1,-1)$. So, the moduli space has two end points. The end points correspond to reducible connections.

Returning to $Z$, the covering transformation preserves the classes $\lambda =(1,1)$ and $(1,-1)$, so the corresponding $\mathop {{SO}}(2)$ connections on $\tilde {Z}$ descend to $\mathop {{SO}}(2)$ connections on $Z$. The moduli space on $Z$ is one dimensional, so must include an arc joining these two points. That is, the arc which is contained in the moduli space of $\tilde {Z}$ consists of invariant connections which descend to $Z$.

Lemma 4.5, together with the description of the $\mathop {{SO}}(3)$ moduli space in last lemma above, gives us a description of the low-dimensional parts of $M(Z,\Sigma )_{*}$.

Proposition 4.7 The $\kappa =0$ part of $M(Z,\Sigma )_{*}$ consists of two points, each of which has monodromy group the cyclic group $\langle \mathbf {i} \rangle \subset \mathop {{SU}}\nolimits (2)$ of order $4$. Under the Klein 4-group action (tensoring by flat line bundles), these are each fixed up to gauge equivalence by the action of tensoring by the line bundle $\xi [-1,-1,1,1]$ (in the obvious notation from above). The two connections are interchanged by tensoring with $\xi [1,-1,-1,-1]$. These two points belong to the moduli spaces $M(Z,\Sigma )_{\epsilon _{1}}$ and $M(Z, \Sigma )_{\epsilon _{2}}$.

The $\kappa =1/4$ part of $M(Z,\Sigma )_{*}$ consists of four arcs, together perhaps with some circles. The Klein 4-group acts transitively on the four arc components of the moduli space. Two of the arcs belong to $M(Z,\Sigma )_{0}$ and two belong to $M(Z,\Sigma )_{\epsilon _{1}+\epsilon _{2}}$.

Proof. From the previous lemma, the $\kappa =0$ part of the moduli space $M(Z,\Sigma )_{*}$ consists of a single orbit of $V_{4}$. From Lemma 4.5, we also learn that the stabilizer of the orbit is the two-element subgroup consisting of the trivial line bundle and the line bundle $\xi [-1,-1,1,1]$. Since the fundamental group of the complement of $\Sigma$ is $\mathbb{Z}/2$, there is no flat $\mathop {{SU}}\nolimits (2)$ bundle on $Z{\setminus} \Sigma$ whose holonomy on the links of $\Sigma$ is conjugate to the element $\mathbf {i}$ of order $4$. So, the flat $\mathop {{SU}}\nolimits (2)$ connection exists only on $Z{\setminus} (\Sigma \cup E_{1} \cup E_{2})$ and must have holonomy $-1$ on the link of exactly one $E_{i}$. These flat connections therefore belong to $M(Z,\Sigma )_{\epsilon _{1}}$ and $M(Z,\Sigma )_{\epsilon _{2}}$.

We now turn to the $\kappa =1/4$ part of the moduli space. The previous lemmas again tell us that the Klein 4-group acts freely and the quotient is a 1-manifold containing a single arc. Therefore, $M(Z,\Sigma )_{*}$ contains four arcs. We are left to determine which of the four parts of $M(Z,\Sigma )_{\nu }$ ($\nu =*$) these belong to. An instanton $[A] \in M(Z,\Sigma )_{*}$ belonging to one of these arcs pulls back to an $\mathop {{SU}}\nolimits (2)$ instanton $[\tilde {A}]$ on

\[ \tilde{Z} {\setminus} \big(\tilde{\Sigma} \cup \tilde{E}_{1} \cup \tilde{E}_{2}\big) \]

with limiting holonomy $-1$ on the links of $\tilde {\Sigma }$. The limiting holonomy on the links of the sphere $\tilde {E}_{i}$ will be $(-1)^{\delta _{i}}$, where $\delta _{i}=1$ or $0$ according to whether $\epsilon _{i}$ appears in $\nu$. Because $[\tilde {E}_{i}]=[\tilde {\Sigma }_{1}]+[\tilde {\Sigma }_{2}]$ in mod-2 homology, we then obtain

\[ w_{2}( \mathrm{ad}(\tilde{A}) ) = (1 + \delta_{1}+\delta_{2})([\tilde{\Sigma}_{1}] + [\tilde{\Sigma}_{2}]). \]

However, the previous lemma tells us that the Stiefel–Whitney class of $[\mathrm {ad}(\tilde {A})]$ is dual to $[\tilde {\Sigma }_{1}]+[\tilde {\Sigma }_{2}]$. Therefore, the possibilities are only $(\delta _{1}, \delta _{2})=(0,0)$ or $(1,1)$. The four arcs therefore belong to the components $M(Z,\Sigma )_{0}$ and $M(Z,\Sigma )_{\epsilon _{1}+\epsilon _{2}}$. Two lie in each, because of the symmetry that arises from the $V_{4}$ action.

Corollary 4.8 If $A$ and $A'$ are the two (abelian) connections which comprise the zero-dimensional part of $M(Z,\Sigma )_{*}$, and if $m_{1}$ and $m_{2}$ are links of the two components of $\Sigma$, oriented so that $A$ has monodromy $\mathbf {i}$ around both links, then the monodromies of $A'$ around $m_{1}$ and $m_{2}$ are $\mathbf {i}$ and $-\mathbf {i}$, up to overall conjugacy.

Let us return now to the pair $(X'_{2}, \Sigma '_{2})$, which we equip with a cylindrical end $\mathbb{R}^{+}\times (S^{3}, U_{2})$. With notation adapted from the discussion of $(Z,\Sigma )$, we examine the moduli space

\[ M(X'_{2}, S'_{2})_{*} \]

on the cylindrical end moduli spaces, with Stiefel–Whitney class $\nu =*$ running over the same four values. The $\mathop {{SU}}\nolimits (2)$ representation variety of $(S^{3}, U_{2})$ is an interval, which we denote by $[-1,1]$, so we have a map

\[ r : M(X'_{2}, S'_{2})_{*} \to [{-}1,1]. \]

From Proposition 4.7 and a stretching argument, we learn that the $\kappa =0$ part $M(X'_{2}, S'_{2})_{*}$ consists of two points which are mapped by $r$ to end points of the interval $[-1,1]$. From Corollary 4.8 we learn that the two points map to opposite ends of the moduli space.

Similarly, we learn that the $\kappa =1/4$ part of $M(X'_{2}, S'_{2})_{*}$ contains four arcs, and that these are each mapped to $[-1,1]$ in such a way that the two end points of each arc map to opposite ends of $[-1,1]$.

Having described these moduli spaces on the cylindrical-end manifold, we now describe how these give rise to the formula in Proposition 4.4 for the case of the finger move. We can break the formula up into:

1. (i) terms coming from the classes $\nu =0$ and $\nu =(\epsilon _{1} + \epsilon _{2})$ on the one hand; and

2. (ii) terms coming from the classes $\nu =\epsilon _{1}$ and $\nu =\epsilon _{2}$

(cf. (38) and (39) above). The first case is that of the four arcs that comprise the $\kappa =1/4$ moduli space. Here the discussion closely mirrors the argument for the finger move in [Reference KronheimerKro97] and [Reference Kronheimer and MrowkaKM11]. Each of the four arcs will contribute a term to the formula having the shape

\[ T_{0}^{x} T_{0}^{y} I^{\sharp}(X, S; \Gamma)_{\omega}, \]

where $x$ and $y$ are curvature integrals for $\mathop {{SO}}(2)$ connections on the components $\Sigma '_{1}$, $\Sigma '_{2}$ of the singular set in the cylindrical-end bifold $(X_{2}',S'_{2})$. The action of the group $V_{4}$ is by tensoring with real line bundles, the effect of which is to change the signs of $x$ and $y$. So, the formula for the four arcs together has the form

\[ \big( T_{0}^{x}T_{0}^{y} + T_{0}^{{-}x}T_{0}^{y} + T_{0}^{x}T_{0}^{{-}y} + T_{0}^{{-}x}T_{0}^{{-}y}\big) I^{\sharp}(X, S; \Gamma)_{\omega}. \]

By symmetry, we have $x=y$ up to sign. So, the formula simplifies to

\[ ( T_{0}^{2x} + T_{0}^{{-}2x}) I^{\sharp}(X, S; \Gamma)_{\omega}. \]

A special case of the finger move is a pair of twist moves, one positive and one negative. So, by comparing this formula to the case of the twist moves, we see that $x=\pm 1$. So, the contribution of the $\kappa =1/4$ moduli spaces to the formula for $I^{\sharp }(X,S^{*};\Gamma )_{\omega }$ is

(40)\begin{equation} ( T_{0}^{2} + T_{0}^{{-}2}) I^{\sharp}(X, S; \Gamma)_{\omega}. \end{equation}

Turning finally to the contributions from the classes $\nu =\epsilon _{1}$ and $\nu =\epsilon _{2}$, we have seen that the $\kappa =0$ moduli spaces

\[ M(X_{2}, S_{2}')_{\epsilon_{1}} \quad\text{and}\quad M(X_{2}, S_{2}')_{\epsilon_{2}} \]

each consist of a single point and these map to the two end points of $[-1,1]$. We can compare this to the moduli space $M(X_{2}, S_{2})$ with $\kappa =0$ (where $(X_{2}, S_{2})$ is now a ball with two disks, equipped with a cylindrical end). For the latter moduli space, the map

\[ r: M(X_{2}, S_{2})_{\kappa=0} \to [{-}1,1] \]

is a homeomorphism. Let us pick points $p$ and $q$ on the two disks and orientations $o_{p}$ and $o_{q}$ nearby. The flat connections on the orbifold $(X_{2}, S_{2})$ are determined by the holonomies around oriented meridians at this point, as in § 3.2, or equivalently by unit vectors $i_{p}$ and $i_{q}$ in the $\mathbb{R}^{3}$ fibers $\mathbb{E}_{p}$ and $\mathbb{E}_{q}$. Under the homeomorphism $r$, the end points of the interval correspond to flat connections with $i_{p}=i_{q}$ and $i_{p}=-i_{q}$ (when we identify $\mathbb{E}_{p}$ with $\mathbb{E}_{q}$ via paths to the base point). If we work with the based moduli space of flat connections on $(X_{2}, S_{2})$, then we instead obtain a map

\[ r: \tilde{M}(X_{2},S_{2})_{\kappa=0} \to [{-}1,1], \]

where the domain is now $S^{2}\times S^{2}$ and the preimage of the end points is the union of the diagonal and anti-diagonal. This is precisely the intersection of $\tilde {M}(X_{2},S_{2})_{\kappa =0}$ with the standard representatives of the two-dimensional cohomology classes $\lambda _{pq}$ and $\lambda '_{pq}$ from (27). If we recall the relation $\lambda _{pq} + \lambda '_{pq}= w_{2}(\mathbb{E}_{p})$ from Lemma 3.6, then we learn that the preimage of the two end points comprises a standard representative for the Poincaré dual of $w_{2}$. From Lemma 3.1 and a stretching argument, it then follows that for the original closed pair $(X,S)$ and the pair $(X,S^{*})$ obtained by the finger move, the contribution to $I^{\sharp }(X,S^{*} ; \Gamma )_{\omega }$ coming from these moduli spaces is $P \, I^{\sharp }(X, S; \Gamma )_{\omega }$. This formula and the terms (40) together give the formula in Proposition 4.4 for the finger move:

(41)\begin{equation} I^{\sharp}(X, S^{*}; \Gamma)_{\omega} = ( P + T_{0}^{2} + T_{0}^{{-}2}) \,I^{\sharp}(X, S ; \Gamma)_{\omega} \end{equation}

or more succinctly

(42)\begin{equation} I^{\sharp}(X, S^{*}; \Gamma)_{\omega} =L \,I^{\sharp}(X, S ; \Gamma)_{\omega}. \end{equation}

Remark As discussed in § 4.3 above, we can choose to change our definition for the blow-ups and use the formula (35). A little extra book-keeping is then required, but the final result needs only slight modification. For the resulting functor $I^{\sharp }_{\xi }$, the statement of Proposition 4.4 is unchanged except for the formula for the factor $L$. We record this as a proposition.

Proposition 4.9 As in Proposition 4.4, let $S^{*}$ be obtained from $S$ by either a positive twist move or a finger move. Let the modified functor $I^{\sharp }_{\xi }$ be defined using the blow-up rule (34). Then we have

\[ I^{\sharp}_{\xi}(X, S^{*} ; \Gamma_{\sigma})_{\omega} = \sigma(L_{\xi})\quad I^{\sharp}_{\xi}(X, S ; \Gamma_{\sigma})_{\omega}, \]

where

\[ L_{\xi} = \xi P + T_{0}^{2} + T_{0}^{{-}2}. \]

For the negative twist move, the map $I^{\sharp }_{\xi }$ is again unchanged:

\[ I^{\sharp}(X, S^{*} ; \Gamma_{\sigma})_{\omega} = I^{\sharp}(X, S ; \Gamma_{\sigma})_{\omega}. \]