The question you're asking is applicable for iOS and Windows but not really for Android.
Android doesn't really have a concept of an application as an object, although there's an Application class. Instead, an app is a loose collection of Activities. There are many good reasons for this state of affairs; for example, it supports fast app switching and easy interaction between Activities of different apps.
The best way to coordinate your "app" so that one Activity doesn't try to do something that's already been done is to use SharedPreferences to store app state. Nearly every other way of doing it is less preferred. Even if the system kills off your entire app, SharedPreferences will maintain the current state. The Application object won't.
Also, Android is based on pausing and resuming. An Activity or activities are created, pause, and resume. They may be destroyed, but that's an extreme case. A corollary to this is that apps should not have an exit button; there's no need for one. I sometimes see apps that have one, but what they're really trying to do is shut down a background Service or process. The best way to do that is to have an affordance that says "Sleep" or similar.