The sidewalk in front of your house is a public-right-of-way, but responsibility for the condition of that sidewalk lies with you, not the Department of Public Works (DPW). If your sidewalk is damaged, cracked or otherwise in need of repair, you must have it fixed. In fact, you could be found liable if a pedestrian sustains an injury on your stretch of concrete –– all the more reason to keep your sidewalk in good shape.
However, before commencing with any work, make sure to call the DPW's Bureau of Street Use and Mapping at (415) 554-5810. The agency will send out an inspector to assess the condition of your sidewalk, suggest the best way to make the necessary repairs, and explain how to obtain the proper permit from the DPW.
To report an unsafe sidewalk, you can call the same Bureau of Street Use & Mapping number above. If the sidewalk in question is the responsibility of the City, the DPW will perform the maintenance. If the sidewalk fronts private property, a DPW inspector will get in touch with the owner and issue him or her a "Notice to Repair" outlining what must be done to bring the sidewalk up to code. If the "Notice to Repair" is ignored, the DPW may step in to do the work and send a bill to the owner for the costs.