Standard constructions:

1. $\gamma_1 * \gamma_2$ juxtaposition of paths $\gamma_1$ and $\gamma_2$
2. $[\gamma]$ the homotopy class of a loop $\gamma$ (with a chosen base point)
3. $G_1\oplus G_2$ direct sum of abelian groups or coproduct of $G_1$ and $G_2$ in AbGr
4. $G_1 * G_2$ Free product of groups or coproduct of $G_1$ and $G_2$ in Gr
5. $X\sqcup Y$ disjoint union of topological spaces or coproduct of $X$ and $Y$ in Top
6. $X\sqcup_f Y$ adjunction space
7. $G_1\times G_2$ direct product of groups $G_1$ and $G_2$
8. $X\times Y$ product of topological spaces $X$ and $Y$
9. $X\vee Y$ wedge of topological spaces $X$ and $Y$
10. $F_k$ free group on $k$ generators or $\mathbb{Z} * \mathbb{Z} \dots *\mathbb{Z}$ ($k$ factors)
11. $[G, G]$ the commutator subgroup of $G$