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Let $X= T \vee \mathbb{C}P^2,$ where $ T$ denotes the 2-dimensional torus.

The task is to compute $\pi_2(X)$ and describe the action of $\pi_1(X)$ on $\pi_2(X)$.

As for the first part, is there any appropriate Serre fibration I could write long exact sequence for? Homotopy groups are known not behave well regarding wedge products. On the other hand, how could on compute homotopy groups of universal cover (Universal cover of $T^2 \vee \mathbb{R}P^2 $) - is it simply connected and thus by Hurewicz theorem: $\pi_2(X)\approx\pi_2(\widetilde{X})\approx H_2(\widetilde{X})$. Secondly, how to describe action of $\pi_1$ on higher homotopy groups? Is not a path-connected space the space where the fundamental group acts trivially on all homotopy groups?

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You're right about the universal cover, so you should try to describe it explicitly to understand its $H_2$.

Then for the action of $\pi_1(X)$, note the following result:

Let $X$ be a nice connected space with universal cover $\tilde X$. Then there is a nice action of $\pi_1(X)$ on $\tilde X$. This induces an action on $\pi_2(\tilde X)$ : with the isomorphism $\pi_2(\tilde X)\cong \pi_2(X)$, this is exactly the action of $\pi_1(X)$ on $\pi_2(X)$

Note that this is not well stated : indeed homotopy groups require basepoints, and the action of $\pi_1(X)$ on $\tilde X$ does not preserve them, so as such, the action on $\pi_2(\tilde X)$ not well-defined.

But $\tilde X$ is simply-connected, so $\pi_2(\tilde X, x)$ and $\pi_2(\tilde X, y)$ are canonically isomorphic, via any path $x\to y$ in $\tilde X$.

Therefore if you're precise you can state this result correctly, and with the correct statement it becomes almost obvious.

But then you can compute this action on $\pi_2(\tilde X)$ via the same action on $H_2(\tilde X)$ by Hurewicz, and with a geometric understanding of $\tilde X$ (and of why its homology is what it is), you'll get a geometric understanding of the action on $H_2(\tilde X)\cong \pi_2(\tilde X)\cong \pi_2(X)$

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