I can't say for sure, especially without knowing where you are and how the jurisdiction of the sidewalk in front of your home is written into town code,
but I have a feeling that if ANYTHING goes wrong, i.e., another local citizen tripping or falling while in proximity to your newly performed work, that you might be
opening a case for liability, in that event. It's one thing to say "if it ain't broke, don't fix it" and another thing to say that you may have no right to do any public
work (even if it's for good), because some municipalities, as well as trouble-making neighbors, might find their pleasure in penalizing you for doing "the right thing".
First, find out which agency is responsible. Next, take a few photos of the sidewalk which will document the deteriorated condition of the sidewalk and lastly, once
you have determined whether you have the right to repair it, then you may set about to fix it.
Gone are the days when we could take it upon ourselves to better our property or enrich our own homes with the assumption that it's "all good", the repairs or
revisions that are done for the betterment of self as well as our neighbors. This is the age of torts and lawsuits, codes and violations, unions and licensed contractors,
many municipalities are more focused in law than in the reality that better neighborhoods bring better neighbors. Town inspectors are finally being told by their superiors,
to go out into the public arena and look a little harder at the things going on in their community, they are often placed in the position of having to justify their salaries and
benefits, by bringing in the revenue that comes from those who improve their property without keeping the proper agency in the picture. Yes, it sounds insane but still, it is
where we have gone, as lowly citizens in a society, over-governed by bloated departments, and those tentacles of government who are so intent upon ruling us, even unto
the ultimate ruination of society.