Cops are looking for a man behind three sex attacks in three hours in downtown Manhattan, a spree that culminated in the rape of a woman in her apartment Thursday morning.

In his most brutal attack, the suspect emerged from a stairwell and pushed his way into the 40-year-old woman’s Lower East Side apartment near the FDR Drive and raped her at around 8:45 a.m, cops said.

He was caught on surveillance video in the building’s lobby at about the time of the sex crime.

That attack came after the man had followed a 33-year-old woman into her James Street apartment, where he attempted to rob and rape her at around 5:40 a.m.

The woman fought him off, and he fled the building.

At 7:40 a.m., the suspect attempted another sexual assault, following a 24-year-old woman into her Broome Street apartment and grabbing her by the shoulders.

She screamed and, again, the attacker fled.

The victim in the final incident was taken to a local hospital, where she was treated and released.

The other two women were not injured in the attempted sexual assaults.

In the surveillance video, the suspect can be seen wearing a black cap and jacket with work boots.

He is believed to be in his mid-20s to mid-30s and stands about 5-foot-10.