Exercises

1. Show that the maps $i_1$ and $i_2$ in definition (23.1) are injective and that the images of $i_1$ and $i_2$ generate $G_1 * G_2$. Hint: Use the universal property with $H = G_1$, $f_1 = i_1$ and $i_2 = 1$.
2. Show that abelianizing a free group on $k$ generators results in a group isomorphic to the direct sum of $k$ copies of $\mathbb{Z}$. Use the fact that the coproduct in AbGr is the direct sum.
3. Is there a surjective group homomorphism from the direct sum $\mathbb{Z}\times \mathbb{Z}$ onto $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$? Prove that if $k$ and $l$ are distinct positive integers, the free group on $k$ generators is not isomorphic to the free group on $l$ generators.
4. Show that $\langle a, c\;\vert\; a^2c^2 = 1\rangle$ is also a presentation of the fundamental group of the Klein's bottle.
5. Construct the push-out for the pair $j_1:C\longrightarrow A_1$ and $j_2:C\longrightarrow A_2$ in the category AbGr?
6. Suppose that $C$ is the trivial group in the definition of push-out in the category Gr, show that the resulting group is the coproduct of the two given groups. What happens in the category AbGr? Describe explicitly the construction of the group specifying the subgroup that is being factored out.

in

Lecture - XXV Adjunction Spaces

in The notion of push-outs in the category Top leads to an important class of spaces known as adjunction spaces. We shall see that most of the important spaces encountered are adjunction spaces. This lecture may be regarded as one on important examples of topological spaces.