# Maxine: Square \(\K\dash\)theory and cut-and-paste manifolds (Thursday, June 16)

> Todo: nice pictures!

:::{.definition title="SK"}
Define
\[
\SK_n = \freemod{\Mfd^n}/\text{diffeo, SK}
.\]
This is a priori only a monoid, but in fact forms a group.
:::

:::{.remark}
By cutting a torus into 4-pieces and regluing into 2 torii, $[T^2] = 2[T^2]$, so $[T^2] = 0$.
Also $[S^2]\inv = [\Sigma_g]$.
:::