Why is there 4 activities missing on your map on saints row the third

There are two reasons.

1) You haven't unlocked them yet. If you haven't played (And completed) all the other activities, doing some will unlock the rest of the activities, not all in one go though.

2) If you are asking this because you haven't got total control of the city yet, you may have not found all of the gang operations. You would have found some randomly when doing missions or when roaming around.

Go to the places in the city that you haven't got control over and search for a group of enemies using the minimap, walk towards them and if it is a gang operation, a fist icon will show in the minimap.

There is one gang operation right in the middle of the city where the people are on top of a very tall building, and that building is slightly outside of the indicated area, to the west a little. There's a good chance this gang operation is one of your four "missing activities".