Generally speaking, unless it is explicitly written that residents of a town are not responsible for repairing a sidewalk, then you can safely assume any cleaning, upkeep, and repairs are entirely on the homeowner.
Often times, even the simplest of repairs may require more work than originally considered. For instance, one of the biggest culprits of busted or raised sidewalks are massive trees lined along the sidewalk. As a result, your city will need to get involved with the removal process. Some towns even have laws against the removal of trees as part of a conservation program, so you’ll certainly want to check with your town to see their stance on that.
Additionally, since the town is getting involved, a permit will be needed, should you wish to have the tree removed.
Before you consider replacing the sidewalk on your own, you may also want to consult the town on any specifics needed to do so. Your best bet would be to find a local mason who is up to speed on the town codes and regulations and has done a few similar jobs along the way. In the grand scheme of things, it’s not all that difficult to mix some cement and slap it down into the walkway, but if it’s not up to code, or if the town finds out you didn’t do it up to code, you could face not only a hefty fine, but also the headache of having to redo the job entirely. So, if you are the sort of person who errs on the side of caution, do it right the first time and connect with a local expert.