macbook air cannot detect TV

First, make sure that the appropriate HDMI input is selected on the tv.

Make sure the adapter and cable are well connected.

Open System Preferences, hold down the option key and click "Detect Displays". Any sign of the mac recognizing the tv?

Also, you can see if the mac detects an external display is connected by seeing if "Arrangement" is one of the buttons at the top of System Preferences->Displays:

How are you connecting to the tv? Which adapter specifically?

Is this one of those usb adapters?

If you could get someone to loan you an adapter, it would be useful to test, as it can isolate the problem.