More on the Bradford killings, with video
Timeline: Bradford prostitute murders
The three women at the centre of the probe
The man was arrested on suspicion of the murders of missing Bradford sex workers Suzanne Blamires, Shelley Armitage and Susan Rushworth, West Yorkshire Police said.
The remains of the woman, who has not yet been identified, were discovered by a member of the public in the River Aire, in Shipley, West Yorkshire, at about 2pm yesterday.
Today, Assistant Chief Constable Jawaid Akhtar said police had been granted extra time, until Thursday evening, to question the 40-year-old man, who was arrested on Monday.
Speaking at the scene of the discovery of the body parts, Mr Akhtar said: "The man is being questioned on suspicion of the murder of Suzanne Blamires, aged 36, who went missing on Friday.
"He has also been arrested and will be questioned on suspicion of the murder of Shelley Armitage, aged 31, who has been missing since Monday April 26, and the murder of Susan Rushworth, aged 43, who has been missing since Monday 22 June 2009."
Mr Akhtar added: "It is a very thorough and painstaking inquiry into three missing women, all of them sex workers, with all the necessary resources and expertise devoted to it."
He said the families of the three women were all being supported by police family liaison officers.
Mr Akhtar was speaking in front of the large white forensics tent police erected by the side of the river.
The remains were found in the water just a few yards from where the busy Otley Road dual carriageway crosses the Aire.
Screens were erected on either side of the bridge and forensic work continued in the small industrial estate where the tents have been put up.
Police divers were at the scene yesterday but left the area today.
A number of officers, including white-suited forensic teams, were still at the scene, behind a car showroom about five miles from Bradford's red light area.
Teams of police, including specially trained sniffer dogs, were also conducting a detailed search of the Chain Street area of Bradford's red light district tonight.
Officers in black uniforms were carrying out systematic searches of streets - stretched across the roads in lines - and other, smaller groups of police were working at locations throughout the Sunbridge Road area.
People living and working in the area said the officers had been examining a number of derelict buildings in the past few weeks and had been searching the area for days.
One householder said: "It's terrible what seems to have happened.
"There are girls on these street every night. Loads of them.
"They're so close together they don't even need mobiles, they just yell to each other.
"It's drugs and prostitutes everywhere round here."
Ms Rushworth was last seen at midday on June 22 last year, after getting off a single-decker bus near her flat on Oak Villas, in the Manningham area.
The grandmother and mother-of-three, who was known as Sue or Susie, suffered from epilepsy and was receiving help for her heroin addiction when she disappeared.
After her disappearance, her 23-year-old son James appealed for information, saying his mother had no reason to go missing.
Mr Rushworth said at the time: "We are all very worried about her. We're a close family and we're not coping well with her disappearance."
Today, neighbours of Ms Rushworth said they feared the worst.
Mario Demski, 42, a worker at Oak Lodge residential home on Oak Villas, said she had visited elderly residents at the home before her disappearance.
He said: "We know that she came here before she went missing and several patients know her but I don't think she has a relative here.
"She lived just over the road. Residents have recognised her face in the paper and been very concerned.
"The police have also spoken to us since she's been missing."
Ms Armitage, who had drug and alcohol problems and was described as "a much-loved daughter and sister", was captured on CCTV on the evening of April 26 and was last seen in Rebecca Street, in Bradford city centre.
Detectives said the disappearance of the 31-year-old, from Bentcliffe Walk, in the Allerton area, was all the more unusual as she had recently bought a puppy which she adored and hated to be parted from for long.
Jan Harrison, who works in a market near Bradford's red-light district, said she would often speak to Ms Armitage and described her as "just a normal girl".
Ms Blamires, whose home on Barkston Walk, in Allerton, is just streets from that of Ms Armitage, has not been seen since Friday last week.
Police tonight said there was no evidence to suggest the investigation into the disappearances of the three women was linked with any other missing person or outstanding murder inquiries after reports that detectives were also looking into three other unsolved cases.
Tonight police were standing guard outside a block of flats on Thornton Road, in Bradford.
Detectives and specialist officers were seen entering Holmfield Court, which is a housing association property.
Officers were checking everyone coming in and out of the converted stone building and a minor road to the side was taped off.