How does the 4th Amendment apply to police entry into a person’s home?

Consider the following Scenario: A Case of ‘Who did it?’

Mary Ellis, a widow who resides with her adult son William, awakens on a Saturday morning and goes to her walk-in closet, where she finds a man whom she recognizes as a neighbor, Clyde Stevens, lying on the floor unresponsive. Mrs. Ellis calls 911. Minutes later, police and EMS personnel arrive. Mr. Stevens is pronounced dead from an apparent stabbing, as he has a large butcher knife protruding from his back. Mrs. Ellis is transported to the hospital for observation, as she is quite distraught.