2. ${\text{Pin}}^{\pm}$-structures and embedded surfaces

For every $n \in \mathbb{N}$, there are two distinct central extensions

(1)\begin{equation} {\text{Pin}}^{\pm}(n) \rightarrow O(n) \end{equation}

which are topologically a disjoint union $\text{Spin}(n) \cup \text{Spin}(n)$. In the orientable setting, it is known that the existence of a Spin-structure on a real vector bundle ξ on a manifold X is equivalent to the vanishing of $w_2(\xi) \in H^2(X,{\mathbb{Z}}_2)$, where w_i is the ith Stiefel-Whitney class of ξ. In the case in which X is 4-dimensional, this equivalent to having an even intersection form, provided that $H^2(X;\mathbb{Z})$ has no 2-torsion [Reference Gompf and Stipsicz12, Corollary 5.7.6]. This can be shown via the Wu formula, which states that for any $a \in H_2(X;{\mathbb{Z}}_2)$ one has

(2)\begin{equation}\langle w_2(X),a\rangle = a^2 \text{ modulo} \ 2\end{equation}

where a ² denotes the algebraic count of self-intersections of a surface in X representing a, see [Reference Gompf and Stipsicz12, Proposition 1.4.18]. In a similar way, there are cohomological obstructions to the existence of ${\text{Pin}}^+$ and ${\text{Pin}}^-$-structures on the tangent bundle of a manifold, namely the vanishing of $w_2(X)$ and of $(w_2+w_1^2)(X)$ respectively, see [Reference Kirby and Taylor17, Lemma 1.3]. Moreover, both ${\text{Pin}}^{+}$ and ${\text{Pin}}^-$-structures (when they do exist) are torsors over $H^1(X,{\mathbb{Z}}_2)$, meaning that there is a free and transitive action of $H^1(X;{\mathbb{Z}}_2)$ over the sets of such geometric structures. However, the Wu formula does not hold when X is non-orientable, but one can still interpret the vanishing of $w_2(X)$ and $(w_2+w_1^2)(X)$ in terms of properties of embedded surfaces. Indeed, for any 4-manifold X and $a \in H_2(X;{\mathbb{Z}}_2)$ there is a (possibly non-orientable) smoothly embedded surface $\Sigma \subset X$ representing a (see [Reference Gompf and Stipsicz12, Remark 1.2.4]) and one can show that

(3)\begin{equation} \langle w_2(X),a\rangle = \chi(\Sigma) +(w_1(\Sigma) \smile w_1(\nu(\Sigma)))([\Sigma])+[\Sigma]^2 \text{ modulo } 2. \end{equation}

Since the tangent bundle of a 4-manifold supports a ${\text{Pin}}^+$-structure if and only if $w_2(X)$ vanishes, this is equivalent to checking that the evaluation (3) is trivial on any homology class $a \in H_2(X;{\mathbb{Z}}_2)$.

On the other hand, the obstruction for ${\text{Pin}}^-$-structures to exist is $(w_2+w_1^2)(X)$ [Reference Kirby and Taylor17, Lemma 1.3] so, in order to understand when such structures exist, we need to compute $w_1^2(X)$ and one can easily show that

(4)\begin{equation} \langle w_1^2(X),[\Sigma] \rangle = w_1^2(\Sigma)+ w_1^2(\nu(\Sigma)). \end{equation}

Example 1 ( $\mathbb{R} \mathbb{P}^4$)

The 4-dimensional real projective space $\mathbb{R} \mathbb{P}^4$ is an example of non-orientable 4-manifold supporting two distinct ${\text{Pin}}^+$-structures but no ${\text{Pin}}^-$-structure. We have that $H_2(\mathbb{R} \mathbb{P}^4;{\mathbb{Z}}_2) \cong {\mathbb{Z}}_2$ is generated by the homology class of $\mathbb{R} \mathbb{P}^2$. Moreover, the tubular neighbourhood $\nu(\mathbb{R} \mathbb{P}^2) \subset \mathbb{R} \mathbb{P}^4$ is diffeomorphic to the twisted 2-disk bundle $D^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ defined as the quotient of $D^2\times S^2$ via the involution

\begin{align*} D^2 \times S^2 \rightarrow D^2 \times S^2 \\ (x,y) \mapsto (\rho_{\pi}(x), -y) \end{align*}

where ρ_π denotes the rotation of S ² of π radians about a fixed axis. One can show that such a quotient is diffeomorphic to

(5)\begin{equation} D^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2 \cong (D^2 \times D^2) \cup_{\varphi} (D^2 \times \text{Mb}) \end{equation}

where Mb denotes the Möbius strip and φ is the map

\begin{equation*} \varphi: D^2 \times S^1 \rightarrow D^2 \times S^1, \quad (x, \theta) \mapsto (\rho_{\theta}(x), \theta) \end{equation*}

and $\rho_{\theta}: D^2 \rightarrow D^2$ denotes the rotation of the 2-disk of angle θ with respect to the origin, see [Reference Akbulut1, Section 0] and [Reference Bais and Torres4, Example 7].

By (3) we have that

\begin{equation*} \langle w_2(\mathbb{R} \mathbb{P}^4), [\mathbb{R} \mathbb{P}^2] \rangle = \chi(\mathbb{R} \mathbb{P})^2+ (w_1(\mathbb{R} \mathbb{P}^2)\cup w_1(\nu (\mathbb{R} \mathbb{P}^2)))([\mathbb{R} \mathbb{P}^2])+[\mathbb{R} \mathbb{P}^2]^2=0 \in {\mathbb{Z}}_2 \end{equation*}

since $\chi(\mathbb{R} \mathbb{P}^2)=1=[\mathbb{R} \mathbb{P}^2]^2$ and $w_1(\nu (\mathbb{R} \mathbb{P}^2))=0$, while (4) implies that

\begin{equation*} \langle w_1(\mathbb{R} \mathbb{P}^4)^2, [\mathbb{R} \mathbb{P}^2] \rangle = w_1^2(\mathbb{R} \mathbb{P}^2) + w_1^2(\nu (\mathbb{R} \mathbb{P}^2))= 1+0=1\in {\mathbb{Z}}_2 \end{equation*}

where $w_1^2(\mathbb{R} \mathbb{P}^2)$ is computed as the self-intersection of a loop in $\mathbb{R} \mathbb{P}^2$ whose ${\mathbb{Z}}_2$-homology class generates $H_1(\mathbb{R} \mathbb{P}^2;{\mathbb{Z}}_2)$.

3. Non-orientable Lefschetz fibrations

Since the conditions 3 and 4 are not immediate to check in the non-orientable setting and depend heavily on the embeddings of the homologically essential surfaces, in the following sections we develop another way to combinatorially understand when a 4-manifold is ${\text{Pin}}^+$ or ${\text{Pin}}^-$ by means of a specific kind of decomposition. To do this, we will need the notion of non-orientable Lefschetz fibration over the 2-disk.

We start by recalling the definition of Lefschetz fibration. Our main reference for this topic in the non-orientable realm is [Reference Miller and Özbağcı19].

Definition 1. Let X be a compact, connected 4-manifold and let B be a compact, connected surface, both with possibly non-empty boundary. A Lefschetz fibration is a smooth submersion

\begin{equation*}f : X \rightarrow B \end{equation*}

away from finitely many points in the interior of B such that each fibre contains at most one critical point and f is a fibre bundle with surface fibre on the complement of the critical values. Moreover, around each critical point, we require f to conform to the local complex model

\begin{equation*}(z_1, z_2) \rightarrow z_1z_2.\end{equation*}

In the following, we will just consider the case in which $B=D^2$ and Σ is the regular fibre. It is possible to show that every Lefschetz fibration

\begin{equation*}f: X \rightarrow D^2\end{equation*}

is obtained from the product one

\begin{equation*}\Sigma \times D^2 \rightarrow D^2\end{equation*}

by gluing 4-dimensional 2-handles along $\Sigma \times \partial D^2$ with framing ±1 with respect to the fibre framing. The attaching curves of such 2-handles are push-offs in distinct fibres of simple closed curves inside Σ, which are called the vanishing cycles of the fibration and are denoted in the following by $c_1, \dots, c_n \subset \Sigma$. Moreover, $\langle w_1(X),[c_i] \rangle =0$ for all $i=1, \dots, n$ and hence the tubular neighbourhood of these attaching regions is necessarily diffeomorphic to a product of S ¹ with the unit interval. We will call two-sided all curves with this property.

4. ${\text{Pin}}^-$-structures on 4-manifolds via Lefschetz fibrations

It is well known that the datum of a Spin-structure over an orientable surface is equivalent to the one of a quadratic enhancement

\begin{equation*} s: H_1(\Sigma;{\mathbb{Z}}_2) \to {\mathbb{Z}}_2\end{equation*}

satisfying the condition

\begin{equation*}s(x+y)=s(x)+s(y)+ x \cdot y\end{equation*}

for all $x,y \in H_1(\Sigma;{\mathbb{Z}}_2)$, where · denotes the ${\mathbb{Z}}_2$-intersection number between cycles, see [Reference Johnson16] and [Reference Stipsicz23]. A geometric interpretation of this algebraic object can be given as follows. The 1-dimensional Spin-cobordism group is $\Omega_1^{\text{Spin}}\cong {\mathbb{Z}}_2$ [Reference Milnor20] and every closed simple curve $\gamma \subset \Sigma$ in a Spin surface inherits a Spin-structure. In particular, if s is the quadratic enhancement associated to the fixed Spin-structure on Σ, then $s([\gamma])=0$ if and only if the induced Spin-structure on γ is the one bounding the unique Spin-structure on the 2-disk.

It is possible to give a similar description for ${\text{Pin}}^-$ structures on (not necessarily orientable) surfaces, as shown by the following result.

Theorem 5 (Kirby-Taylor, [Reference Kirby and Taylor17])

There is a 1:1 correspondence between ${\text{Pin}}^-$-structures on a surface Σ and quadratic enhancements

\begin{equation*} q^{-}: H_1(\Sigma; {\mathbb{Z}}_2) \to {\mathbb{Z}}_4\end{equation*}

with the property that

\begin{equation*}q^{-}(x+y)=q^{-}(x)+q^{-}(y)+2 \ x \cdot y\end{equation*}

for any $x,y \in H_1(\Sigma; {\mathbb{Z}}_2)$.

In particular, if $\gamma \subset \Sigma$ is a simple closed curve, then $q^{-}([\gamma])$ is even if and only if γ is two-sided and this is the case whenever γ is a vanishing cycle for a Lefschetz fibration. Moreover, recall that the 1-dimensional ${\text{Pin}}^-$-cobordism group is $\Omega_1^{{\text{Pin}}^-}\cong {\mathbb{Z}}_2$ (see [Reference Anderson, Brown and Peterson3, Theorem 5.1] and [Reference Kirby and Taylor17, Section 0]) and every closed simple curve $\gamma \subset \Sigma$ in a ${\text{Pin}}^-$ surface inherits a ${\text{Pin}}^-$ structure. We have that $q^{-}([\gamma])=0$ if and only if γ inherits the bounding ${\text{Pin}}^-$-structure.

In order to prove Theorem 1, we will need the following Lemma.

Lemma 1. Let $f: X \rightarrow D^2$ be a Lefschetz fibration with regular fibre Σ and vanishing cycles $c_1, \dots, c_n \subset \Sigma$. There is a natural one to one correspondence between the set of ${\text{Pin}}^-$-structures on X and the set of quadratic enhancements

\begin{equation*}q^-:H_1(\Sigma;{\mathbb{Z}}_2) \rightarrow {\mathbb{Z}}_4\end{equation*}

such that $q^-(x+y)=q^-(x)+q^-(y)+2x\cdot y$ for all $x, y \in H_1(\Sigma;{\mathbb{Z}}_2)$ and $q^-([c_i])=2$ for all $i=1, \dots, n$. Moreover, such correspondence is equivariant with respect to the free and transitive action of $H^1(X;{\mathbb{Z}}_2)$.

Proof. The existence of a ${\text{Pin}}^-$-structure on X is equivalent to the existence of a ${\text{Pin}}^-$-structure on Σ that extends to the 2-handles. This follows from the fact that any ${\text{Pin}}^-$-structure on X induces by restriction a ${\text{Pin}}^-$ structure on a (trivial) tubular neighbourhood $\nu(\Sigma)\cong \Sigma \times D^2$ and all the ${\text{Pin}}^-$-structures on $\Sigma \times D^2$ are pull-backs of the ones on Σ via the projection map $\Sigma \times D^2 \rightarrow \Sigma$, see also the proof of [Reference Stipsicz23, Theorem 1.1]. Since such 2-handles are attached to $\Sigma \times \partial D^2$ with relative odd framing, this corresponds to a ${\text{Pin}}^-$-structure on Σ restricting to the non-bounding one on every vanishing cycle. The conclusion follows from Theorem 5.

The free and transitive action of $H^1(X;{\mathbb{Z}}_2)$ on the set of ${\text{Pin}}^-$-structures on X can be seen as follows. In [Reference Kirby and Taylor17, Section 3], it is shown that $H^1(\Sigma;{\mathbb{Z}}_2)$ acts on the set of ${\text{Pin}}^-$-structures of the surface Σ by

(6)\begin{equation} q^{-}_{\gamma}(x)=q^{-}(x)+2 \cdot \gamma(x) \end{equation}

for all $\gamma \in H^1(\Sigma;{\mathbb{Z}}_2)$ and $x \in H_1(\Sigma;{\mathbb{Z}}_2)$. In particular, ${\text{Pin}}^-$-structures on Σ equivariantly correspond to quadratic enhancements and the action (6) is well defined on $H^1(X;{\mathbb{Z}}_2)\subset H^1(\Sigma;{\mathbb{Z}}_2)\cong H^1(X;{\mathbb{Z}}_2)\oplus K$, where $K \subset H^1(\Sigma;{\mathbb{Z}}_2)$ is the kernel of the map induced by the inclusion $\Sigma \subset X$.

In light of this fact, it is possible to characterize the existence of ${\text{Pin}}^-$-structures on non-orientable Lefschetz fibrations over the 2-disk in terms of the ${\mathbb{Z}}_2$-homology classes of the vanishing cycles.

Proof of Theorem 1

By Lemma 1, X does not support any ${\text{Pin}}^-$-structure if and only if it is not possible to build a map

\begin{equation*}q^-: H_1(\Sigma; {\mathbb{Z}}_2) \rightarrow {\mathbb{Z}}_4 \end{equation*}

such that $q^-(x+y)=q^-(x)+q^-(y)+2x\cdot y$ for all $x,y \in H_1(\Sigma;{\mathbb{Z}}_2)$ and such that $q^+([c_i])=2$ for all $i=1, \dots, n$. The remaining part of the proof is essentially the same as the proof of [Reference Stipsicz23, Theorem 1.1]. We will now sketch it for the convenience of the reader. Let $V\subset H_1(\Sigma;{\mathbb{Z}}_2)$ be the vector subspace generated by the classes of the vanishing cycles. Notice that any given enhancement $q^-$ on V can be extended to the whole $H_1(\Sigma;{\mathbb{Z}}_2)$ by setting $q^-=0$ on a basis $u_1, \dots, u_k$ of an orthogonal vector subspace and using the relation (6) to define it on the vectors of the form $\sum_j u_{i_j}+v$, where $v \in V$. In particular, in order to define a ${\text{Pin}}^-$-structure on X, one needs to find a quadratic enhancement $q^-: V \rightarrow {\mathbb{Z}}_4$ which takes value 2 on the class of any vanishing cycle. Let $v_1, \dots, v_r$ be a basis of V represented by vanishing cycles and let $V_i=\langle v_1, \dots, v_i \rangle$ for $i=1, \dots, r$. We then want to define $q^-$ inductively on each V_i as in the proof of [Reference Stipsicz23, Theorem 1.1], setting $q^-(v_i)=2$ at each step. One then notices that things go wrong if and only if there are k classes $v_{i_1}, \dots, v_{i_k}$ such that their sum $v_0=\sum_{j=1}^k v_{i_j} \in H_1(\Sigma;{\mathbb{Z}}_2)$ is again represented by a vanishing cycle and $q^{-}(v_0)\neq 2 \in {\mathbb{Z}}_4$. This implies that $q^-(v_0)=0 \in {\mathbb{Z}}_4$, being v ₀ is represented by a curve with trivial tubular neighbourhood. Since by construction we had that $q^-(v_{i_j})=2$ for $j=1, \dots, k$, using (6) the condition $q^-(q_0)=0 \in Z_4$ translates into

\begin{equation*}q^-(v_0)=\sum_{j=1}^kq^-(v_{i_j})+ 2\cdot \Big( \sum _{1 \leq i_n \leq i_m \leq k} v_{i_n} \cdot v_{i_m} \Big)=2\cdot \Big(k+\sum _{1 \leq i_n \leq i_m \leq k} v_{i_n} \cdot v_{i_m}\Big)=0 \in {\mathbb{Z}}_4.\end{equation*}

The conclusion then follows by noticing that $2 \cdot x =0 \in {\mathbb{Z}}_4$ if and only if $x=0 \in {\mathbb{Z}}_2$.

Note that, if we restrict to orientable surfaces, there is a natural map

\begin{equation*}\Omega_1^{\text{Spin} }\to \Omega_1^{{\text{Pin}}^- }\end{equation*}

giving a group isomorphism, see [Reference Kirby and Taylor17, Theorem 2.1]. At the level of the associated quadratic forms, the enhancement

\begin{equation*}s: H_1(\Sigma;{\mathbb{Z}}_2)\to {\mathbb{Z}}_2\end{equation*}

corresponds to

\begin{equation*}q^{-}: H_1(\Sigma;{\mathbb{Z}}_2)\to {\mathbb{Z}}_4\end{equation*}

given by

\begin{equation*}q^{-}(x) = 2 \cdot s(x) \end{equation*}

for every $x \in H_1(\Sigma;{\mathbb{Z}}_2)$, where $2 \cdot$ denotes the inclusion ${\mathbb{Z}}_2 \subset {\mathbb{Z}}_4$. In particular, the condition on the vanishing cycles we found in Theorem 1 coincides with the one in [Reference Stipsicz23, Theorem 1.1] and this is due to the fact that, when restricting to orientable Lesfschetz fibrations over D ², being Spin coincides with being ${\text{Pin}}^-$.

5. ${\text{Pin}}^+$-structures on Lefschetz fibrations over the 2-disk

As a consequence of [Reference Degtyarev and Finashin7, Theorem A], there is a canonical affine bijective correspondence between ${\text{Pin}}^+$-structures on a surface Σ and the set of maps

(7)\begin{equation} q^{+}: H_1(\Sigma;{\mathbb{Z}}_4) \to {\mathbb{Z}}_2 \end{equation}

with the property that

(8)\begin{equation} q^{+}(x+y)=q^{+}(x)+q^{+}(y)+x \cdot y \end{equation}

for every $x, y \in H_1(\Sigma;{\mathbb{Z}}_4)$, where $x \cdot y\in {\mathbb{Z}}_2$ denotes the algebraic intersection modulo 2 between representatives of the ${\mathbb{Z}}_2$-reductions of the classes $x,y \in H_1(\Sigma;{\mathbb{Z}}_4)$.

Recall that the 1-dimensional ${\text{Pin}}^+$-cobordism group is

\begin{equation*}\Omega_1^{{\text{Pin}}^+}\cong \{0\}\end{equation*}

see [Reference Kirby and Taylor17, Section 3]. However, S ¹ has two distinct ${\text{Pin}}^+$-structures. These correspond to two different trivializations of the direct sum $TS^1 \oplus \varepsilon$ of the tangent bundle of S ¹ with a trivial line bundle, where the first one is given by the restriction to S ¹ of a trivialization of the tangent bundle $T D^2$ of the 2-disk and the second one comes from the Lie group framing of S ¹. In particular, one can show that such structures have the property that they respectively bound the 2-disk and the Möbius strip, see [Reference Kirby and Taylor17, Section 1]. A close look at the construction of $q^{+}$ starting from a ${\text{Pin}}^+$-structure on Σ shows that $q^{+}([\gamma])=0$ if the induced ${\text{Pin}}^+$-structure on γ is the one bounding a 2-disk, while $q^{+}([\gamma])=1$ if it is the one bounding the Möbius strip, see [Reference Kirby and Taylor17]. Note that, given any two maps $q^{+}_1, q^{+}_2$ corresponding to ${\text{Pin}}^+$-structures on Σ, their difference can be regarded as an element

\begin{equation*}q^{+}_1 - q^{+}_2 \in\text{Hom}(H_1(\Sigma;{\mathbb{Z}}_4),{\mathbb{Z}}_2).\end{equation*}

The proof of Theorem 2 is based on the following Lemma.

Lemma 2. Let $f: X \rightarrow D^2$ be a Lefschetz fibration with regular fibre Σ and vanishing cycles $c_1, \dots, c_n \subset \Sigma$. There is a natural bijection between the set of ${\text{Pin}}^+$-structures on X and the set of quadratic enhancements

\begin{equation*}q^+:H_1(\Sigma;{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2\end{equation*}

such that $q^+(x+y)=q^+(x)+q^+(y)+x \cdot y$ for all $x, y \in H_1(\Sigma; {\mathbb{Z}}_4)$ and $q^+([c_i])=1$ for all $i=1, \dots, n$. Moreover, such correspondence is equivariant with respect to the free and transitive action of $H^1(X;{\mathbb{Z}}_2)$.

Proof. As is the ${\text{Pin}}^-$ case, the existence of a ${\text{Pin}}^+$-structure on X is equivalent to the existence of a ${\text{Pin}}^+$-structure on Σ that extends to the 2-handles. Since such 2-handles are attached to $\Sigma \times \partial D^2$ with relative odd framing, this corresponds to a ${\text{Pin}}^+$-structure on Σ restricting to the one bounding the Möbius strip on every vanishing cycle. The conclusion follows from [Reference Degtyarev and Finashin7].

In the ${\text{Pin}}^+$-case, $H^1(\Sigma;{\mathbb{Z}}_2)$ acts on the set of ${\text{Pin}}^+$-structures of the surface Σ by

(9)\begin{equation} q^{+}_{\gamma}(x)=q^{+}(x)+ \gamma(x) \end{equation}

for all $\gamma \in H^1(\Sigma;{\mathbb{Z}}_2)$ and $x \in H_1(\Sigma;{\mathbb{Z}}_4)$, see [Reference Degtyarev and Finashin7]. In particular, ${\text{Pin}}^+$-structures on Σ equivariantly correspond to quadratic enhancements and the action (9) is well defined on $H^1(X;{\mathbb{Z}}_2)\subset H^1(\Sigma;{\mathbb{Z}}_2)\cong H^1(X;{\mathbb{Z}}_2)\oplus K$, where $K \subset H^1(\Sigma;{\mathbb{Z}}_2)$ is the kernel of the map induced by the inclusion $\Sigma \subset X$.

Let $f: X \to D^2$ be a Lefschetz fibration over the 2-disk with surface fibre Σ. Let $c_1, \dots, c_n \subset \Sigma$ be its vanishing cycles. Note that one can find simple closed curves $e_1, \dots, e_g$ inducing a basis of $H_1(\Sigma;{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2^g$. From now on, we will assume that Σ has a ${\text{Pin}}^+$-structure, since this is a necessary condition for X to be a ${\text{Pin}}^+$-manifold. In particular, this means that Σ is not a closed non-orientable surface of odd Euler characteristic. Let

(10)\begin{equation} q^{+}_0: H_1(\Sigma;{\mathbb{Z}}_4) \to {\mathbb{Z}}_2 \end{equation}

be the quadratic enhancement associated to a fixed ${\text{Pin}}^+$-structure on Σ and let $c_j=\sum_{i=1}^r c_{ji}[e_i].$ This gives the setting of Theorem 2, of which we now provide a proof.

Proof of Theorem 2

Lemma 2 implies that X has a ${\text{Pin}}^+$-structure if and only if there is a map

\begin{equation*}q^+: H_1(\Sigma,{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2\end{equation*}

such that $q^+(x+y)=q^+(x)+q^+(y)+x \cdot y$ for all $x, y \in H_1(\Sigma; {\mathbb{Z}}_4)$ and $q^+([c_i])=1$ for all $i=1, \dots, n$. In particular, every such $q^+$ is necessarily of the form $q^{+}=q^{+}_0+l$ for some linear map $l=\sum_{i=1}^r \lambda_i [e^i]$, where $[e^i]\in H^1(\Sigma;{\mathbb{Z}}_2)$ denotes the dual element to $[e_i]\in H_1(\Sigma;{\mathbb{Z}}_2)$. In particular, this reduces the problem to finding $l= \sum_{i=1}^r \lambda_i e^i$ such that for every $j=1, \dots, n$ we have

\begin{equation*} q^{+}([c_j])=q^{+}_0([c_j])+l([c_j])=q_0^+([c_j])+\sum_{i=1}^r c_{ji} \lambda_i=1.\end{equation*}

The conclusion follows from Rouché-Capelli’s Theorem [Reference Shafarevich and Remizov22, Theorem 2.38].

6. Some examples

In this section, we show how to apply the results proven so far. In particular, starting from the handlebody decomposition of some small non-orientable 4-manifold M, we endow the union of handles up to index 2 with the structure of a Lefschetz fibration over the 2-disk, following the procedure explained in the proof of [Reference Miller and Özbağcı19, Theorem 1.1]. At this point, we apply Theorem 1 and Theorem 2 to understand whether or not M admits a ${\text{Pin}}^+$ or a ${\text{Pin}}^-$-structure, describing the action of $H^1(M;{\mathbb{Z}}_2)$ in terms of the action of a subgroup of $H^1(\Sigma;{\mathbb{Z}}_2)$ on the quadratic enhancements on the non-singular fibre Σ of the Lefschetz fibration. We refer the reader to [Reference Akbulut1, Section 1.5] for the conventions and backgound on Kirby diagrams of non-orientable 4-manifolds, see also [Reference Akbulut, Gordon and Kirby2], [Reference Bais and Torres4, Section 2.1], [Reference César de Sá6] and [Reference Miller and Naylor18]. The procedure we describe in this section applies to any closed non-orientable 4-manifold represented by means of a Kirby diagram.

Example 2 ( $\mathbb{R} \mathbb{P}^4$)

Figure 1 represents a Kirby diagram for the standard handlebody decomposition of $\mathbb{R} \mathbb{P}^4$ with a single k-handle for each $k=0, \dots, 4$. In particular, the union of all handles up to index 2 corresponds to a tubular neighbourhood $\nu(\mathbb{R} \mathbb{P}^2) \cong D^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ of $\mathbb{R} \mathbb{P}^2$ (see [Reference Akbulut, Gordon and Kirby2, Section 0] and [Reference Bais and Torres4, Section 2.5]) and $\mathbb{R} \mathbb{P}^4$ is the result of attaching a 3-handle and a 4-handle to this handlebody. Recall that 3-handles and 4-handles need not be drawn by [Reference César de Sá6] and [Reference Miller and Naylor18], see [Reference Bais and Torres4, Example 7].

Figure 1. A Kirby diagram of $\mathbb{R} \mathbb{P}^4$.

The union of handles up to index 2 can be seen as a Lefschetz fibration over the 2-disk with the Möbius band Mb as fibre and a single vanishing cycle c ₁, given by the projection of the $(1,0)$-framed 2-handle, see Figure 2.

Figure 2. The Lefschetz fibration associated to the 2-handlebody of $\mathbb{R} \mathbb{P}^4$.

A ${\text{Pin}}^+$-structure on Mb extending to $\mathbb{R} \mathbb{P}^4$ is given by the quadratic enhancement

(11)\begin{equation} q^{+}:H_1(\text{Mb};{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2 \end{equation}

defined by the condition $q^{+}([e_1])=1$, where $[e_1]$ is the generator of $H_1(\text{Mb};{\mathbb{Z}}_4) \cong {\mathbb{Z}}_4$ represented by the core of the band. Indeed, $[c_1]=2[e_1]$ and hence

\begin{equation*}q^+([c_1])=q^{+}(2[e_1])= q^{+}([e_1])+q^{+}([e_1])+e_1^2=1 \in {\mathbb{Z}}_2.\end{equation*}

The other ${\text{Pin}}^+$-structure

\begin{equation*}q^{+}_x(y)=q^{+}(y)+x \cdot y \quad \text{for all } y \in H_1(\text{Mb}; {\mathbb{Z}}_4)\end{equation*}

is obtained by acting on $q^{+}$ by the cohomology class dual to $[e_1]$, where $x \cdot y$ indicates the intersection number between x and the ${\mathbb{Z}}_2$-reduction of y. Note that the condition $q^{+}_x([c_1])=1$ is still satisfied.

On the other hand, if $\mathbb{R} \mathbb{P}^4$ supported a ${\text{Pin}}^-$-structure, then there would be a quadratic enhancement

\begin{equation*}q^{-}:H_1(\text{Mb};{\mathbb{Z}}_2) \rightarrow {\mathbb{Z}}_4\end{equation*}

satisfying $q^{-}([c_1])=2$. But this is impossible, since it would imply that

\begin{equation*}2=q^{-}([c_1])=q^{-}(2[e_1])=q^{-}(0)=0 \in {\mathbb{Z}}_4.\end{equation*}

Example 3 ( $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$)

The non-orientable 4-manifold $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ is obtained as a quotient of $S^2\times S^2$ under the orientation-reversing involution

\begin{align*} \iota : S^2 \times S^2 \rightarrow S^2 \times S^2 \\ (x,y) \mapsto (-x, \rho_{\pi}(y)) \end{align*}

where $\rho_{\theta}: S^2 \rightarrow S^2$ is the rotation of S ² of angle θ about a fixed axis for any $\theta \in S^1$, see [Reference Hillman15, Chapter 12]. Moreover, it can also be seen as the result of a Gluck twist on the product $S^2 \times \mathbb{R} \mathbb{P}^2$ along a 2-sphere fibre, see [Reference Bais and Torres4, Lemma 1]. In particular, there is a decomposition

\begin{equation*} S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2= (S^2 \times D^2) \cup_{\varphi} (S^2 \times \text{Mb})\end{equation*}

where the gluing map is

\begin{align*} \varphi: S^1 \times S^1 \rightarrow S^2 \times S^2 \\ (x, \theta ) \mapsto (\rho_{\theta}(x), \theta). \end{align*}

A Kirby diagram of this manifold is given in Figure 3, see [Reference Akbulut, Gordon and Kirby2, Section 0], [Reference Bais and Torres4, Figure 1], [Reference Miller and Naylor18] and [Reference Torres25, Section 4.1]. It is obtained by considering $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ as the double of $D^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$. Figure 4 represents the projection of the attaching circles of the 2-handles onto the page of the trivial Lefschetz fibration given by the union of handles up to index one. In order to endow the 2-handlebody of $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ with the structure of a Lefschetz fibration over D ², we need to stabilize the page and add vanishing cycles as in Figure 5, so that the attaching spheres of the 2-handles can be isotoped into different pages and have the right framing. This is Harer’s trick and is explained in the non-orientable setting in [Reference Miller and Özbağcı19, Proof of Theorem 1.1], see also [Reference Harer14] and [Reference Etnyre and Fuller8, Theorem 2.1]. At the level of Kirby diagrams, this corresponds to adding canceling pairs of 1- and 2-handles.

Figure 3. A Kirby diagram of $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$.

Figure 4. Lefschetz fibration associated to the 2-handlebody of $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$.

Figure 5. The 2-handlebody of $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ as a Lefschetz fibration over D ².

In Figure 6 we fix a set of simple closed loops $e_1, \dots, e_6$ inducing a homology basis of the page Σ of this new fibration. The quadratic enhancement

\begin{equation*}q^{+}:H_1(\Sigma;{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2\end{equation*}

Figure 6. Curves representing a basis of $H_1(\Sigma).$

defined by the condition $q^{+}([e_i])=1$ for every $i=1, \dots, 6$ has the property that $q^{+}([c_i])=1$ for all the vanishing cycles $c_1, \dots, c_7$ and hence it determines a ${\text{Pin}}^+$-structure on $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$. The other ${\text{Pin}}^+$-structure can be found by acting on $q^{+}$ with $[e_1]\in H_1(\Sigma;{\mathbb{Z}}_2)$.

On the other hand, $S^2 \mathbin{\widetilde{\smash{\times}}} \mathbb{R} \mathbb{P}^2$ does not support a ${\text{Pin}}^-$-structure. Indeed, if this was the case, we could find a quadratic enhancement

\begin{equation*}q^{-}: H_1(\Sigma;{\mathbb{Z}}_2) \rightarrow {\mathbb{Z}}_4\end{equation*}

such that $q^{-}([e_i])=2$ for all $i=1, \dots, 7$ and this would imply that

\begin{align*}2&=q^{-}([c_1])=q^{-}([e_2+e_6])=q^{-}([e_2])+q^{-}([e_6])=q^{-}([c_2])+q^{-}([c_6])=2+2\\ &=0\in {\mathbb{Z}}_4\end{align*}

a contradiction.

Example 4 ( $S^2 \times \mathbb{R} \mathbb{P}^2$)

A Kirby diagram of the product 4-manifold $S^2 \times \mathbb{R} \mathbb{P}^2$ is given in Figure 7, see [Reference Akbulut, Gordon and Kirby2, Section 0], [Reference Bais and Torres4, Figure 1], [Reference Miller and Naylor18] and [Reference Torres25, Section 4.1]. It is obtained by considering $S^2 \times \mathbb{R} \mathbb{P}^2$ as the double of $D^2 \times \mathbb{R} \mathbb{P}^2$. Figure 8 shows the projection of the attaching circles of the 2-handles onto the page of the trivial Lefschetz fibration given by the union of handles up to index one.

Figure 7. A Kirby diagram of $S^2 \times \mathbb{R} \mathbb{P}^2$.

Figure 8. Lefschetz fibration associated to the 2-handlebody of $S^2 \times \mathbb{R} \mathbb{P}^2$.

We use Harer’s trick again (see [Reference Miller and Özbağcı19, Proof of Theorem 1.1], [Reference Harer14] and [Reference Etnyre and Fuller8, Theorem 2.1]) and show that the union of the handles up to index 2 in the fixed decomposition of $S^2 \times \mathbb{R} \mathbb{P}^2$ is given by the surface Σ and by the vanishing cycles $c_1, \dots, c_8$ in Figure 9. Note that, with respect to the case of $S^2 \widetilde \times \mathbb{R} \mathbb{P}^2$, we need to stabilize the page one additional time in order to adjust the framing of the red coloured 2-handle.

Figure 9. The 2-handlebody of $S^2 \times \mathbb{R} \mathbb{P}^2$ as a Lefschetz fibration over D ².

In Figure 10 we again fix a set of loops $e_1, \dots, e_7 \subset \Sigma$ inducing a first homology basis of the page.

Figure 10. Curves representing a basis of $H_1(\Sigma)$.

The quadratic enhancement

\begin{equation*}q^{-}:H_1(\Sigma;{\mathbb{Z}}_2) \rightarrow {\mathbb{Z}}_4\end{equation*}

defined by the condition $q^{-}([e_i])=2$ for every $i=1, \dots, 7$ has the property that $q^{-}([c_i])=2$ for all the vanishing cycles $c_1, \dots, c_8$ and hence it determines a ${\text{Pin}}^-$-structure on $S^2 \times \mathbb{R} \mathbb{P}^2$. The other ${\text{Pin}}^-$-structure can be found by acting on $q^{-}$ with $[e_1]\in H_1(\Sigma;{\mathbb{Z}}_2)$.

On the other hand, $S^2 \times \mathbb{R} \mathbb{P}^2$ can not support a ${\text{Pin}}^+$-structure. Indeed, if this was the case we could find a quadratic enhancement

\begin{equation*}q^{+}: H_1(\Sigma;{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2\end{equation*}

such that $q^{+}([c_i])=1$ for all $i=1, \dots, 8$ and this would imply that

\begin{align*} 1=q^{+}([c_1])=q^{+}([2e_1+e_2+e_6+e_7])=\\ 2q^{+}([e_1])+e_1^2+q^{+}([e_2])+q^{+}([e_6])+q^{+}([e_7])=\\ 1+q^{+}([c_2])+q^{+}([c_6])+q^{+}([c_7])=1+1+1+1=0 \in {\mathbb{Z}}_2 \end{align*}

a contradiction.

7. ${\text{Pin}}^{\pm}$-structures on Lefschetz fibrations over S ²

In this section, we provide a proof of Theorem 4.

8. ${\text{Pin}}^+$ and ${\text{Pin}}^-$-structures on 3-manifolds

The following well-known fact can be easily proven by adapting to the ${\text{Pin}}^-$ case the one of [Reference Stipsicz23, Theorem 1.4], see [Reference Kirby and Taylor17].

On the other hand, it is not true that any closed 3-manifold admits a ${\text{Pin}}^+$-structure, since the condition $w_2(M)=0$ is not always satisfied in the non-orientable setting. $\mathbb{R} \mathbb{P}^2 \times S^1 \operatorname{\#} N$ for any 3-manifold N is such an example. However, we now show that it is still possible to check this condition by considering a handlebody decomposition of M.

Theorem 7. Let $M=H \cup H'$ be a closed non-orientable 3-manifold obtained by adding g 2-handles and a 3-handle to a genus-g non-orientable handlebody H and let $q_0^+: H_1(\partial H;{\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2$ be a quadratic enhancement corresponding to a ${\text{Pin}}^+$ structure on $\partial H$. Let $a_1, \dots, a_g \in H_1(\partial H;{\mathbb{Z}}_4)$ and $b_1, \dots, b_g \in H_1(\partial H;{\mathbb{Z}}_4)$ be the homology classes of the attaching circles of the 2-handles and of the belt circles of the 1-handles of H respectively and set

\begin{equation*}q^{+}_0([a_j])=\alpha_j \quad \text{and} \quad q^{+}_0([b_j])=\beta_j\end{equation*}

for $j=1, \dots, g$. If for a fixed basis $e_1, \dots, e_{2g}$ of $H_1(\partial H;{\mathbb{Z}}_2)$ we have

\begin{equation*}\tilde a_j=\sum_{i=1}^{2g} a_{j,i} e_i \quad \text{and} \quad \tilde b_j=\sum_{i=1}^{2g} b_{j,i} e_i\end{equation*}

for $j=1, \dots, g$, where $\tilde a_j$ and $\tilde b_j$ are the ${\mathbb{Z}}_2$-reduction of a_j and b_j respectively, then M has a ${\text{Pin}}^+$-structure if and only if

\begin{equation*}\text{rank}(C|A)=\text{rank} (C)\end{equation*}

where C and A are respectively the $2g \times 2g$ matrix and the column vector

(12)\begin{equation} C= \begin{pmatrix} a_{1,1} & \dots & a_{1,2g}\\ \vdots & \ddots & \vdots\\ a_{g,1} & \dots & a_{g,2g} \\ b_{1,1} & \dots & b_{1,2g}\\ \vdots & \ddots & \vdots \\ b_{2g,1} & \dots & b_{g,2g} \end{pmatrix} , \qquad A= \begin{pmatrix} \alpha_1\\ \vdots\\ \alpha_g\\ \beta_1\\ \vdots\\ \beta_g \end{pmatrix} . \end{equation}

Proof. There is a ${\text{Pin}}^+$-structure on M if and only if the non-orientable closed surface $\partial H$ has a ${\text{Pin}}^+$-structure which extends to the attaching circles of the 2-handles and to the belt spheres of the 1-handles in H. In particular, this is equivalent to check whether or not there exists a map

(13)\begin{equation} q^{+}: H_1(\partial H; {\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2 \end{equation}

satisfying the condition $q^+(x+y)=q^+(x)+q^+(y)+x \cdot y$ for all $x,y \in H_1(\partial H;{\mathbb{Z}}_4)$ and which vanishes on $a_1, \dots, a_g$ and on $b_1, \dots, b_g$.

If we fix a quadratic enhancement

(14)\begin{equation} q^{+}_0: H_1(\partial H; {\mathbb{Z}}_4) \rightarrow {\mathbb{Z}}_2 \end{equation}

corresponding to a ${\text{Pin}}^+$-structure on $\partial H$, then all the other ${\text{Pin}}^+$-structures on $\partial H$ will be of the form $q^{+}=q^{+}_0+l$ for some linear map $l \in \text{Hom}(H_1(\partial H;{\mathbb{Z}}_4), {\mathbb{Z}}_2)$. It follows that M supports a ${\text{Pin}}^+$-structure if and only if there is $l=\sum_{i=1}^{2g} x_i e^i\in \text{Hom}(H_1(\partial H;{\mathbb{Z}}_4),{\mathbb{Z}}_2)$ such that

\begin{equation*}\sum_{i=1}^{2g} a_{j,i}x_i=\alpha_j \quad \text{and} \quad \sum_{i=1}^{2g}b_{j,i}x_i=\beta_j\end{equation*}

in ${\mathbb{Z}}_2$ for $j=1, \dots, g$, where $e^1, \dots, e^{2g}$ is the dual basis to $e_1, \dots, e_{2g} \in H_1(\partial H; {\mathbb{Z}}_2)$. The conclusion follows from Rouché–Capelli’s theorem.

9. Interpretation of ${\text{Pin}}^+$-structures à la Milnor

Kirby and Taylor showed in [Reference Kirby and Taylor17] that the sets of ${\text{Pin}}^+$ and ${\text{Pin}}^-$-structures on a fixed vector bundle ξ are in one to one correspondence with Spin-structures on $\xi \oplus 3 \cdot \text{det}(\xi)$ and $\xi \oplus \text{det}(\xi)$ respectively. In this section, we remark that for ${\text{Pin}}^+$-structures one can get another equivalent characterization, which recalls the following one of Spin-structures by Milnor.

Recall that $x\in H^1(M;{\mathbb{Z}}_2)$ acts on a trivialization of $\xi|_{M^1}$ by flipping the framing of any loop $\gamma \subset M^1$ such that $\langle x, [\gamma] \rangle =1 \in {\mathbb{Z}}_2$.

In a similar fashion, we prove the following result. We remark that this follows essentially from the discussion in [Reference Kirby and Taylor17].

We remark that the definition of the action of $H^1(M;{\mathbb{Z}}_2)$ on the set of homotopy classes of linearly independent sections of $\xi|_{M^1}$ is completely analogous to the one described after the statement of Theorem 8.

Proof. The vanishing of $w_2(\xi)$ is a necessary and sufficient condition for the existence of both a $(k-1)$-tuple of linearly independent sections of $\xi|_{M^2}$ and a ${\text{Pin}}^+$-structure on ξ. We now suppose that $w_2(\xi)$ is trivial and we endow $\text{det}(\xi)$ with its canonical ${\text{Pin}}^+$-structure, see [Reference Kirby and Taylor17, Addendum to 1.2]. Let $v_1, \dots, v_{k-1}$ be a $(k-1)$-tuple of linearly independent sections of $\xi|_{M^1}$ extending to $\xi|_{M^2}$. We are now going to associate to such a $(k-1)$-tuple a ${\text{Pin}}^+$-structure on $\xi|_{M^2}$. This will uniquely determine a ${\text{Pin}}^+$-structure on ξ, as in the case of Spin-structures. Using $v_1, \dots, v_{k-1}$ one can define a vector bundle isomorphism

(15)\begin{equation} \xi|_{M^2} \cong \varepsilon^{k-1} \oplus \text{det}(\xi) \end{equation}

where $\varepsilon^{k-1}$ denotes the trivial rank- $(k-1)$ real vector bundle. The associated ${\text{Pin}}^+$-structure on $\xi|_{M^2}$ is given by the pull-back via (15) of the ${\text{Pin}}^+$-structure on $\varepsilon^{k-1} \oplus \text{det}(\xi)$ given by the direct sum of the trivial Spin-structure on $\varepsilon^{k-1}$ with the fixed ${\text{Pin}}^+$-structure on $\text{det}(\xi)$. It is now easy to verify that this correspondence satisfies all the desired properties.