For years before Jordan Neely, a mentally ill homeless man, was killed in the subway, the city had its eye on him.

He was on a list informally known as the Top 50, a roster of people in a city of eight million who stand out for the severity of their troubles and their resistance to accepting help. The list is overseen by a task force of city agency workers and social-service nonprofits; when homeless-outreach workers see someone in the subway who is on the list, they are supposed to notify the city and try to get that person to a shelter.

Despite that, and an open arrest warrant, Mr. Neely was out on his own on May 1, when he began ranting at passengers. A Marine veteran, Daniel Penny, grabbed him and choked him to death; Mr. Penny has now been charged with manslaughter.

In the wake of Mr. Neely’s death, the administration of Mayor Eric Adams has been criticized by advocates for homeless people and left-leaning political opponents who say the killing highlights deep problems in the city’s support systems for homeless people and those with mental illness.

At a news conference on Thursday, Councilwoman Pierina Sanchez, referring to Mr. Neely’s presence on the list, said: “Our city knew exactly who Jordan was, where he was and what his history was. And yet we failed him.”

But as officials describe it, the task force and the Top 50 list were formed precisely for the people whom the system had failed time and again. The death of Mr. Neely, 30, who had been homeless for years, also shows the limits of the tools the task force has at its disposal and the difficulty of keeping track of people who are transient and elusive, let alone getting them to accept help.

In a speech this week, Mr. Adams called the group that maintains the list “the guiding force” behind the city’s efforts to help people like Mr. Neely “stabilize and heal from the ravages of homelessness and long-term, untreated psychosis.”

The goal of the list is to connect disparate bureaucracies across a vast city, in which a group of people with intense needs regularly interacts with hospital personnel, street social workers and police officers who do not regularly interact with each other.

The people on the list are among the city’s “most entrenched and chronic patients,” the mayor’s senior adviser for severe mental illness, Brian Stettin, said in an interview, and are discussed at weekly meetings of the task force.

The group that keeps the list, known formally as the Coordinated Behavioral Health Task Force, consists of workers from across city government, including the departments of Health, Homeless Services and Hospitals, along with representatives of the nonprofits that the city contracts with to try to connect homeless people to shelter and services, a process known as outreach.

At the weekly meetings, Mr. Stettin said, task force members exchange updates on the people on the list — “what their current needs seem to be” and in some cases “how their conditions have changed to the point where we have to start thinking about different ways we can approach their cases.”

Top 50 is a bit of a misnomer. The list does not have a fixed number of people on it, and there are actually two lists — one for people who typically stay in the subways and one for people who stay in the streets. People can be taken off the list for any number of reasons, including moving into housing or going to jail.

Mr. Neely was on the subway list, according to an employee of the Bowery Residents’ Committee, a group that has the city contract to do outreach in the subways.

Outreach workers in the subways are supposed to be familiar with all the names on the subway list, said the Bowery Residents’ Committee employee, who spoke anonymously because they were not authorized to publicly discuss the matter.

When they encounter someone who is on it, the employee said, they are supposed to notify the Department of Homeless Services, which can arrange transport to a type of shelter known as a safe haven. Safe havens have fewer rules and restrictions than barrackslike group shelters and are often more appealing to the homeless people who typically avoid shelters.

For people on the street version of the list, which in early May included about 65 people, being on it can get someone badly needed services more quickly, said Juan Rivera, the outreach director for BronxWorks, which has the city contract to do street outreach in the Bronx.

Mr. Rivera described a man who had been staying at a BronxWorks safe haven, broken some windows in a rage and left for the streets. BronxWorks had him added to the Top 50 list, which allowed him to jump up the wait list to become a client of a street squad of clinicians known as an Intensive Mobile Treatment team.

Working with the mobile treatment team, the man regained stability and has moved into permanent housing, Mr. Rivera said.

“He’s still connected with his team, and he’s doing really well,” he said.

As for Mr. Neely, he had been a fixture for years in the subway system — first in his teens and 20s as a gifted Michael Jackson impersonator who captivated commuters with his fluent moonwalking, and later, as he tumbled into mental illness and drug abuse, as a disheveled and sporadically violent man who racked up repeated arrests and trips to hospitals. Mr. Neely was on the list in 2019, when it was launched, and remained on it until his death, according to the Bowery Residents’ Committee employee.

In February, Mr. Neely, who had been in jail on an assault charge for punching a 67-year-old woman and breaking several bones in her face, was released to a residential treatment program, under a plea deal that required him to avoid trouble for 15 months, stay on antipsychotic medication and not abuse drugs.

Two weeks later, he walked out of the facility and did not return, and the arrest warrant was issued.

In March, Mr. Neely was approached by homeless-outreach workers at a subway station in Manhattan. He was neatly dressed and calm and accepted a ride to a shelter in the Bronx where he spent the night, according to outreach records shared with The New York Times.

But on April 8, when outreach workers found him at an end-of-the-line station in Coney Island, Mr. Neely, wearing dirty clothes riddled with burn holes, exposed himself and urinated inside a subway car, according to the notes shared with The Times.

Outreach workers, whose job requires them to win the trust of people who seek to avoid contact with the authorities, typically do not check for warrants, but they summoned the police, who ejected him from the station.

The police were also apparently unaware of the warrant. A program launched in 2019 in which the police did warrant checks on people caught violating transit-system rules was abandoned during the pandemic, after criticism that it was criminalizing homelessness.

The workers in Coney Island learned only the next day that the person they had met was a man on the Top 50 list, case notes show.

A note later filed by an outreach worker about the encounter reads prophetically: “Due to client’s aggressive behavior, he could be a harm to others or himself if left untreated and not assessed by a mental health professional.”

Under a directive issued by Mr. Adams last fall, people who are in such a severe state of psychological crisis that they are a danger to themselves or to others are supposed to be taken to a hospital for evaluation, involuntarily if necessary.

End-of-the-line subway outreach teams typically include nurses who work for the city and are trained to do field assessments of people and have them transported to hospitals. Outreach notes do not indicate whether a nurse evaluated Mr. Neely at Coney Island.

Mr. Neely had been hospitalized involuntarily in the past, but does not appear to have been during the last months of his life.

Lauren McCarthy contributed reporting.