Can I withhold rent ifmy landlord has not fixinga cockroach infestation?

The answer is *maybe*--though speak with an attorney and follow his/her advice before doing anything. Every lease has added to it by the law what's known as the "implied warranty of habitability." This is a requirement that the premises be fit for their intended purpose, and conditions that make an apartment or home, for example, not fit to live in--like a severe cockroch infestation--may violate the warranty. If the warranty is violated, the tenant may be able to seek monetary compensation, an order from the court requiring the condition to be corrected, or to break the lease without penalty. Another option is to pay for the tenant to pay for the repair, extermination, etc. him- or herself, then deduct the cost from the rent. This last option can be tricky: you have to give the landlord proper written notice and a  chance to correc the situation; then you can only withhold an amount equal to the cost to repair. You cannot generally withhold the rent to force the landlord to act.

So you do have options, but you want to exercise them correctly; failure to do so can result in you being liable to the landlord instead. That is why you should consult with an attorney before acting. If you can't afford one, try contacting Legal Services. Good luck.