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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 $\mathbb{S^1} \times \mathbb{S^1}$.

I have an idea how to do this, but I don't know how to add in the details. Construct the torus T from W by attaching a 2-cell. Then, that 2 cell by construction gives us a homotopy from identity to $\alpha\beta\alpha^{-1}\beta^{-1}$. How can I use this idea to prove the result ?

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    I'll take a wild swing here and say you and @TheKEMO are [probably in the same class](http://math.stackexchange.com/questions/2138208/commuting-elements-in-the-fundmental-group). – 2017-02-10
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    Probably haha yeah. – 2017-02-10
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    Good to know that I am not the only one struggling with this question. – 2017-02-10
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    @DanRust do you have an idea about this question ? – 2017-02-10
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    Sorry on the move atm so can't write an answer. As always with these questions, draw a picture and draw the relevant diagram of composition of maps. – 2017-02-10

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