Yes, that is the bearing housing. The shaft is a round solid piece of steel inside it. The pulley is attached to the shaft, not the bearing housing. Here is a drawing of the assembly, from the illustrated parts list:
![[Linked Image]](https://www.outdoorking-forum.com.au/forum/uploads/usergals/2012/10/full-2772-8349-husqvarna_cutter_bearing_assembly.png)
Item 13 is the shaft, Item 14 is the bearing housing. Item 15 is a ball bearing. There is another ball bearing already on Item 13 - looks as if you can't buy it separately from the shaft.
Can you take some pictures of the damaged bearing housing please, both from above and below the deck? I need to understand exactly which piece has broken, and how. I also need to verify that there is no damage to the deck.