Actually, for just the lane-by-lane mowing this could theoretically work without odometry although the lanes would probably not match perfectly side-by-side - as you can see in the picture, the 'a little bit forward' into the next lane would be not always the same length, and so it might be possible that the lanes don't overlap etc.
However, another important feature is localization and mapping: after perimeter installation, the mower has to create the map by driving on the perimeter. The map will be only precise if we know the distance the mower drives, so we need odometry. Odometry will be used later for localization too. When the mower drives on the perimeter, it can localize itself on the perimeter just by the help of the map.
Localization on perimeter (and within perimeter - not shown here):
Yes, you can use IR-Led and a Wheel witch break the light. Or you use magnet and reed-switch.. But both systems can have problems.. I use IR-Led.. Every wheel count exactly.. But i got still Errors from Ordometrie.. I don't know.. He drive only 5 Meter and stopp with ordo- error..