I want to calculate the second homotopy group of the blow-up of the unit cotangent disk bundle of a closed surface $\Sigma$, i.e $\pi\_2\left(D^\*T\Sigma\#\overline{\mathbb{C}P^2}\right).$ Actually, I want to understand whether these symplectic manifolds are monotone. If yes, I want to find a closed monotone Lagrangian submanifold in it. 



Homotopy groups of the blow-ups and monotone symplectic manifolds
Write $X$ for the manifold in question (i.e. the symplectic blowup of the unit disc bundle in the cotangent bundle of a surface $\Sigma$).


If you're interested just in monotonicity, what's relevant isn't so much $\pi\_2(X)$ as the image of $\pi\_2(X)$ under the Hurewicz map to $H\_2(X;\mathbb{Z})$, which is quite a bit simpler--it has rank either one or two (generated at most by the zero section and the exceptional sphere, depending on what $\Sigma$ is). Now $\omega$ evaluates as zero on the zero section and is positive on the exceptional sphere, while the first Chern class is zero on the zero section and $1$ on the exceptional sphere. Thus $[\omega]$ and $c\_1$ are positively proportional as elements of $H^2\left(X;R\right)$,
so $X$ is always monotone, regardless of $\Sigma$.


Regarding monotone Lagrangian submanifolds, the obvious candidate is the zero section $\Sigma$. Since $\pi\_1(\Sigma)\to \pi\_1(X)$ is an isomorphism, all classes in the relative $\pi\_2$ come from the absolute $\pi\_2$, so since the ambient manifold is monotone it directly follows that $\Sigma$ is a monotone Lagrangian submanifold.


When $\Sigma$ is $S^2$, there's another interesting example of a monotone Lagrangian $L\subset D^\*TS^2$ discussed in [this paper](https://arxiv.org/abs/math/0608356) of Albers and Frauenfelder. If you carefully choose the size of the blowup (so that the proportionality constant on the exceptional sphere is the same as the monotonicity constant for $L$), then it should be possible to arrange for $L$ to lift to a monotone Lagrangian torus in the blowup. (However for a generic size blowup this probably won't work.)


