Yes, you can sue the city if you fall on the sidewalk. The City must have "actual notice" of the defect in the sidewalk, and a claim must be filed with 90 days of the occurrence. It is only worthwhile to sue the City for a fall on the sidewalk is you have sustained a very serious injury. This is because litigation against the City is expensive, and the investment is not merited unless there are enough "damages" to collect.
If you suspect that your sports injury may have been caused by negligence or recklessness, a free consultation with a sports injuries lawyer will help you sort this out. At Sports Injuries Lawyer, Wittenstein & Wittenstein, we've been helping injured people for more than 60 years. Please don't hesitate to call if you think you might have a claim.