Compute $\pi_2(S^2 \vee S^2)$

Question

As far as I know, there are two ways to calculate higher homotopy groups. One way is if we have a fibration then we get a long exact sequence in homotopy. The other is if we know a space is $(n-1)$-connected, then by Hurewicz Theorem, $\pi_n \cong H_n$.

I know $H_2(S^2 \vee S^2)=\mathbb{Z} \oplus \mathbb{Z}$ and $\pi_1(S^2 \vee S^2)$ is trivial as the pushout of $\pi_1(S^2) \leftarrow \pi_1(\ast) \rightarrow \pi_1(S^2)$ (using van Kampen's Theorem).

Is it true that a wedge of $n$-spheres is $(n-1)$-connected? If so, why? If this is the case, Hurewicz applies.

Also, is there any fibration involving $S^2 \vee S^2$? I thought about the Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$, but intuitively I doubt that this produces another fibration $S^1 \vee S^1 \rightarrow S^3 \vee S^3 \rightarrow S^2 \vee S^2$.

Any help you can provide would be appreciated!

Answer 1

I think the question meant to ask about $\pi_2(S^2 \vee S^2)$. Since $\pi_0$ and $\pi_1$ vanish, the Hurewicz map gives an isomorphism $\pi_2(S^2 \vee S^2) \to H_2(S^2 \vee S^2) = \mathbb{Z}^2.$

Answer 2

Any wedge of $n$-spheres is indeed $(n-1)$-connected. One way to see this is to give $\vee_{i=1}^kS^n$ a CW structure with one $0$-cell and $k$ $n$-cells. If $f\colon S^i\to \vee_{i=1}^kS^n$ is any pointed map then by cellular approximation $f$ is pointed-homotopic to a cellular map $\tilde{f}$, and if $i < n$ then $\tilde{f}$ must be constant.