This was sparked because I wanted to compute $\pi_2(Sym^2(\Sigma_2))$ via $Sym^2(\Sigma_2)\approx \mathbb{T}^4$# $\bar{\mathbb{C}P}^2$.
We know how to compute $\pi_1$ of $M$ # $N$ via van-Kampen's theorem. But what about higher homotopy groups? I looked in the literature and google without luck, and so I am wondering if no such procedure exists. Are there any results for calculating $\pi_n$ of connected sums?
There was mention of "higher van Kampen theorem"… has this actually been used to do such computations? I'd be interested in references if not just examples.
Best Answer
The 2nd homotopy group of a connect sum is fairly reasonable to compute. $\pi_i X$ is isomorphic to $\pi_i \tilde X$ provided $i \geq 2$ and $\tilde X$ indicates any covering space of $X$, so we might as well take the universal cover. By the Hurewicz theorem, $\pi_2 \tilde X$ is isomorphic to $H_2 \tilde X$. In the case of a connect sum, the universal cover has a very nice description (take disjoint unions of the universal covers of the punctured manifolds and glue them together appropriately).
Since $\mathbb CP^2$ is simply connected this is a fairly easy thing to compute. The universal cover looks like $\mathbb R^4$ with a $\mathbb CP^2$ summand at every integer lattice point. So,
$$\pi_2 ((S^1)^4 \# \mathbb CP^2) \simeq \bigoplus_{\pi_1 T^4} \pi_2 \mathbb CP^2$$
i.e. a direct sum over $\mathbb Z^4$ of copies of the integers, i.e. $\mathbb Z[t_1^\pm, t_2^\pm, t_3^\pm, t_4^\pm]$ a laurent polynomial ring in four variables. $\pi_1$ acts by multiplication by units in the Laurent polynomial ring.
Higher homotopy groups in general can be fairly painful to compute but $\pi_2$ is usually quite reasonable, like this case.