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Please guys how can anyone show me the proof of the following:

For every space $(X,x)$, show that $[\alpha], [\beta] \in \pi_1(X,x)$ commute if and only if the map $$f: (\mathbb{S^1} \vee \mathbb{S^1}, \star) \rightarrow (X,x)$$ defined by $$ f(s,t)= \alpha (s)\,\,\text {if}\,\, t = \star$$ and $$f(s,t)= \beta (t)\,\,\text {if}\,\, s = \star$$ has a continuous extension over $\mathbb{S^1} \times \mathbb{S^1}$.

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    Well...that's a just plain gorgeous problem. Do you know the fundamental group of $S^1 \times S^1$? – 2017-02-10
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    Yes I do. Its $\mathbb{Z} \times \mathbb{Z}$ – 2017-02-10
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    Commuting gives you a based homotopy from $\alpha\beta$ to $\beta\alpha$. The proof is then essentially just pinching down the constant vertical edges (the edges corresponding to the constant map at the base point), and inverting the process for the converse. – 2017-02-10

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