Have dry cough and chest tightness (been 3 weeks now), no fever/headache. it started after a few times that i went swimming. is it chlorine allergy?

Possible: It has been reported that swimming in chlorinated water actually increases the risk in developing asthma. This is ironic given that swimming is likely the best sport for people with asthma. I suspect that you may have a case of cough-variant asthma or exercise-induced bronchospasm. Check with your doctor.

No: You cannot be allergic to chlorine. It does not have a high enough molecular weight to be the cause of an allergic reaction. Chlorine is, however, an irritant and can cause the symptoms that you describe when exposed to a high enough concentration.