$user->data
should have what you want.
// snip
elseif ( in_array( 'map_user', $user->roles ) )
return home_url( "/mapping/{$user->data->user_login}/" );
// end snip
Add if (!is_wp_error($user)) wp_die(var_dump($user));
to the top of your function to see what you have to work with. Caution: This will break things. (It throws a notice as well) It is for debugging only.