Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

Question

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates nontrivially on the image $G \subset \pi_2(L)$ of the boundary homomorphism $\pi_3(M,L) \to \pi_2(L)$.

Note that such $L$ must have dimension at least $4$, since any $L$ of dimension at most $3$ is $\text{Pin}^-$, which means in particular that $w_2(TL)$ vanishes on $G$. More generally, if $L$ is so-called relatively $\text{Pin}^\pm$, then $w_2(TL)$ vanishes on $G$ as well ($L$ is called relatively $\text{Pin}^+$ if $w_2(TL)$ is in the image of the restriction morphism $H^2(M;\mathbb Z_2) \to H^2(L;\mathbb Z_2)$; it is relatively $\text{Pin}^-$ if the same is true of $w_2(TL) + w_1^2(TL)$). Indeed, suppose $L$ is relatively $\text{Pin}^+$. Then $w_2(TL)|_G$ factors through $\pi_2(L) \to \pi_2(M)$ and therefore it vanishes since of course $\pi_3(M,L) \to \pi_2(L) \to \pi_2(M)$ is zero. The same argument works if $L$ is relatively $\text{Pin}^-$ since $w_1^2(TL)$ vanishes on spheres in $L$.

Lastly, I would really like to obtain such examples where $L$ is moreover assumed to be monotone with minimal Maslov at least $2$.

Answer 1

Audin, Lalonde and Polterovich have a construction (Corollary 1.2.5 in this book) which produces a Maslov-two monotone Lagrangian embedding of $M^n \times T^m$ into $\mathbb{C}^{n+m}$ for any $m \ge 1$, given that $M^n$ admits a Lagrangian immersion into $\mathbb{C}^n$. The latter condition is equivalent to $TM^n \otimes \mathbb{C}$ being a trivial unitary bundle. All we have to do is thus to find our favourite non-spin manifold with trivial complexified tangent bundle. I would go for $M=\mathbb{C}P^2 \sharp \overline{\mathbb{C}P^2}$.