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I have two problems that I don't know how to do )=.

i) Let X path connected. Let $x_0 , x_1 \in X$. Show that $\pi _1 \left( {X,x_0 } \right)$ is abelian iff for every $\alpha, \beta$ paths from $x_0$ to $x_1$ , we have $\widehat\alpha = \widehat\beta$.

Where the hat denotes the homomorphism: $$ \eqalign{ & \widehat\alpha :\pi _1 \left( {X,x_0 } \right) \to \pi _1 \left( {X,x_1 } \right) \cr & \widehat\alpha \left( {\left[ f \right]} \right) = \left[ \alpha \right]^{ - 1} \left[ f \right]\left[ \alpha \right] \cr} $$

I did the side where the group is abelian, but the other I could not )=.

ii) The other is a property that I read in wikipedia. If $X$ and $Y$ are path connected, then $$ \pi _1 \left( {X\times Y} \right) \cong \pi _1 \left( {X\,} \right) \times \pi _1 \left( Y \right) $$ I'm not sure if this nice property is easy to show. )=

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    *(ii)* is a proposition in Hatcher's *Algebraic Topology.*2012-02-23

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