JK Secondary Air Injection is an emissions system that pumps air into the exhaust system at the proper time to help the catalytic converter function more efficiently. A common failure on that truck is that the pump has sucked up water/snow/ice and has failed. This will also blow the fuse(s).
The fuses are in a little fuse block stuck down between the fender and the battery. A black cover over the block, a little hard to see. First check those fuses, i would you will find a blown fuse. If so, you will have to replace the A.I.R. Pump. There is also a service bulletin for this problem. It involves installing new pump, and hose assembly to prevent water from being sucked into it
2003 Chevy Trailblazer 4.2L inline 6 cylinder. Battery replacement. I tried pulling the HVACB and HVAC1 fuses. Left for a minute and started the engine left for 2 minutes. Turned off engine. Turned it back on and tried to see if it would work and it didn't. Dash vents will not work. Manual system.
JK: I would disconnect the negative battery cable for 10 minutes, then do the procedure you have tried again. If that does not help, you may have a problem in the a/c control head. Replacing battery can sometimes do bad things to electronics, especially if not done properly. Re-learning the door positions can also be done with the dealer scanner. This re-home is better than just removing power from the system. And, you may end up there anyway to have the system properly diagnosed
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