Let $g$ and $n$ be positive integers such that $3g-3 + n > 0$ Let $\mathcal{M}_{g,n}$ the moduli stack of genus $g$ nodal curves with $n$ marked points. There are two obvious families of maps
$$\mathcal{M}{g,n+1} \mathcal{M}_{g,n+1} \rightarrow \mathcal{M}{}{g,n}$$mathcal{M}_g{}_n$$and identifying two marked points$$\mathcal{M}_{g_1,n_1} \times \mathcal{M}_{g_2,}{}_{n_2} \rightarrow \mathcal{M}_{g_1 + g_2,}{}_{n_1 + n_2 - 2}$$This system constitutes the so-called Grothendieck-Teichmüller tower. It is indeed intricate and in my opinion, also beautiful. Moreover, it is a conjecture of Grotehndieck that its automorphism group is naturally isomorphic to the absolute Galois group over \mathbb{Q}. 2 edited body; added 2 characters in body Let g and n be positive integers such that 3g-3 + n > 0 Let \mathcal{M}_{g,n} the moduli stack of genus g nodal curves with n marked points. There are two obvious families of maps forgetting a point$$\mathcal{M}_{g,n+1} $\mathcal{M}{g,n+1} \rightarrow \mathcal{M}_g{}_n$$mathcal{M}{}{g,n}$$ and identifying two marked points $$\mathcal{M}_{g_1,n_1} \times \mathcal{M}_{g_2,}{}_{n_2} \rightarrow \mathcal{M}_{g_1 + g_2,}{}_{n_1 + n_2 - 2}$$ This system constitutes the so-called Grothendieck-Teichmüller tower. It is indeed intricate and in my opinion, also beautiful. Moreover, it is a conjecture of Grotehndieck that its automorphism group is naturally isomorphic to the absolute Galois group over$\mathbb{Q}$. 1 [made Community Wiki] Let$g$and$n$be positive integers such that$3g-3 + n > 0$Let$\mathcal{M}_{g,n}$the moduli stack of genus$g$nodal curves with$n$marked points. There are two obvious families of maps forgetting a point $$\mathcal{M}_{g,n+1} \rightarrow \mathcal{M}_g{}_n$$ and identifying two marked points $$\mathcal{M}_{g_1,n_1} \times \mathcal{M}_{g_2,}{}_{n_2} \rightarrow \mathcal{M}_{g_1 + g_2,}{}_{n_1 + n_2 - 2}$$ This system constitutes the so-called Grothendieck-Teichmüller tower. It is indeed intricate and in my opinion, also beautiful. Moreover, it is a conjecture of Grotehndieck that its automorphism group is naturally isomorphic to the absolute Galois group over$\mathbb{Q}\$.