Capt. Lawrence Jackmin said authorities questioned her husband, Blazej Kot, but have not charged him.
Kot, 24, is a doctoral candidate in information services at Cornell. Police took him into custody late Tuesday night after a park policeman on routine patrol noticed his car in a closed parking lot and found Kot covered in blood.
Kot drove off and led police on a five-mile chase before crashing into a stand of trees. He was found with an apparent self-inflicted injury and police had to forcibly take an "edged" weapon from him, Jackmin said.
Police then went to Kot's apartment and found a fire had been set inside. Firefighters put out the blaze in about 15 minutes.
Kot was in stable condition after emergency surgery at an area hospital, where he remained under guard Thursday, Jackmin said.
When police could not locate Coffey, they began sweeping the area for her Wednesday morning. Jackmin said authorities searched for about an hour when they learned two hikers had found a woman's body on a park trail about 400 yards from the apartment.
Jackmin declined to discuss a possible motive.
The couple were married last fall, he said.
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