Gardaí have identified a person of interest as part of the investigation into the murder of Ashling Murphy.
Ms Murphy, a 23-year-old primary school teacher, was attacked and killed on the banks of the Grand Canal in Tullamore, Co Offaly on Wednesday afternoon.
Gardaí are waiting to interview a man who is currently in hospital.
They say they are making "significant progress" in their investigation and expect developments in the next few days.
RTÉ understands that gardaí are carrying out searches at properties in Tullamore and Dublin as part of the investigation into the murder of Ms Murphy.
Earlier today, gardaí said they are particularly interested in knowing more about a Falcon Storm mountain bike with straight handlebars and distinctive yellow/ green front forks.
They also appealed to anyone who was in the Cappincur/Canal walk before 4pm on Wednesday to contact them.
RTÉ News has also learned that another woman reported she had been followed along the Canal Bank at Cappincur two hours before Ms Murphy was murdered.
Last night, a 40-year-old man was released from garda custody and declared no longer to be a suspect in the case.
The man had been arrested by gardaí within an hour of the discovery of the body of Ashling Murphy.
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He had been questioned for seven hours on Wednesday evening before he rested overnight and 15 hours yesterday.
RTÉ News understands the man consistently denied any involvement in the murder during garda interviews.
Detectives continued to interview witnesses and gather forensic evidence, including DNA, and as a result of their inquiries, they discovered yesterday evening that the man could not have been involved in the murder.
The man's solicitor has said that he has had his "life ruined".
Donal Farrelly, who represented the man during his two days of questioning, condemned those who had tried to identify him on social media.
This morning, gardaí renewed their appeal for information as they continue their investigation.
Ms Murphy taught first class at Durrow National School. She left there after work on Wednesday and went to exercise along the banks of the canal at Cappincur in Tullamore.
She was attacked and killed within the hour and died by strangulation.
Gardaí appealed for anyone with information to contact Tullamore Garda Station on 057-9327600, the Garda Confidential Line on 1800-666-111, or any garda station.
Minister for Justice Helen McEntee said the priority is that every effort is made to ensure the person who killed Ms Murphy is brought to justice.
Speaking on Midlands 103 FM, the minister said she spoke with Garda Commissioner Drew Harris this morning and is making sure that gardaí have whatever they need to carry out this investigation.
"Anything that can be done will be done and every resource will be provided."
Minister McEntee said she knows that people are "upset and shocked" and want to know that the right person is brought to justice.