Gizmo, once again you made me crack up...but what if it's not a maple tree?
I've dealt with a few of those situations before, and in each case I solved it differently, because of the different conditions. In my city, the sidewalks are the city's property to maintain. The city gets sued for injuries and pays out millions in settlements rather than fix the pavements - but this is the American way.
It looks like removing the broken concrete, grinding or cutting the roots and pouring new concrete sections is the best solution. It lasts more than a few years.
I have a big city tree in front of a house that provides terrific shade for parked cars when the sun is blazing - why kill it? The sidewalk is a lwasuit waiting to happen and calls to the city are ignored. The city warns that anything done to the tree which is not to code (drilling, copper pipes, copper nails, even trimming) will result in heavy fines...
If your neighbor is fed up with your tree, can he just bring a chain saw and cut it down or damage it? of course no. Well the same way with the city tree.