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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.

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Serj and Mike, sorry for the confusion by my previous wrong answer, it is uneditable, so I deleted it. –  Dmitri Oct 17 '10 at 10:09
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2 Answers

up vote 2 down vote accepted

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 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.)

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Thank you Mike. While struggling with monotonicity via $\pi_2$ I also realized that it actually depends only on spherical homology classes. Thank you for the reference. I am familiar with this torus too. I just wanted to work with monotone Lagrangians that lie in the interior of a convex compact symplectic manifold. I think another candidate for $L$ is the Clifford torus inside a small 4-ball (seating in some Darboux chart) blown-up in the origin –  Serj Oct 17 '10 at 14:35
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I don't know how to comment, so here is a comment on Dmitri's answer:

I think that if $\Sigma$ is not $S^2$ $\pi_2$ is bigger than $\mathbb{Z}$. For $\mathbb{R}P^2$ it is $\mathbb{Z}^3$ and for others it is a free abelian group generated by elements in $\pi_1(\Sigma)$ (each of those elements gives a copy of $\overline{\mathbb{C}P^2}$ in the universal cover).

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Many thanks for the answer. Of course, I can look on the universal cover of the blow-up, which has isomorphic higher homotopy groups. –  Serj Oct 17 '10 at 7:25
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