This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly appreciated. Thanks for your time.
Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?
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algebraic-topology
homotopy-theory
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7For $n>1$, the $n$th homotopy group of a space is the same as the $n$th homotopy group of the universal cover. – 2012-04-11
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4And the $\pi_2$ of the cover is isomorphic to its $H_2$, which is easier to compute. – 2012-04-11
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0(Well, since the two are isomorphic, they are equally easy to compute...) – 2012-04-11
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0Here is an explicit calculation and answer: https://math.stackexchange.com/questions/1675337/calculate-pi-2s2-vee-s1 – 2017-04-16