Er ... so far as I was aware, the push notifications come in via the cell communication signals (like SMS) and add a mini icon (e.g. number of emails waiting) to the application icon on the homescreen. you're only going to use up more battery once you load the application to deal with that status update.
SMS signaling is not used these days for push... it's not reliable and more importantly, it costs the user.
Push requires keeping a communication path open between the phone and server. To keep it open, comms must occur every ten to thirty minutes.
This is done by sending a TCP/IP request with a very long timeout from the phone to the server. If a notification needs to be sent, the server simply replies to the phone right then. If the timeout occurs first, then the server replies negatively and the phone sends another request.
The upshot is, push requires constant data comms, and that requires power.