WASHINGTON (AP) — Just how healthy is the U.S. job market?

Despite steady hiring and falling unemployment, the question has provoked sharp debate and considerable uncertainty on the eve of the September jobs report.

Will millions without jobs who aren’t looking for one eventually start looking?

Why aren’t companies filling more of their openings?

Why can many people find only part-time work?

Much of the uncertainty flows from a big question: Does today’s 6.1 percent unemployment rate, far below the 10 percent it hit in 2009, mean the job market is near full health? Or does the unemployment rate overstate the improvement?

The answers, whenever they come, could play a key role in when the Federal Reserve decides to finally raise interest rates. “We’re kind of grasping at straws,” says Peter Cappelli, an economist at the Wharton Business School. “We’ve never seen a labor market quite like this.”

The government’s September jobs report, coming Friday, may shed some light on these questions. But it won’t settle them.

In the midst of a national campaign season, jobs and unemployment remain near the top of voters’ list of most important issues, according to the latest Associated Press-GfK poll. Only the economy overall and terrorism or national security were significantly more likely to be cited by people who are expected to vote.

Here are five mysteries about the job market:

— How many people who have stopped looking for work — or never started — will start looking if the economy improves further?

This, perhaps more than any other question, has confounded economists. Since the Great Recession began in late 2007, the proportion of adults either working or seeking work has sunk from 66 percent to 62.8 percent — a 35-year low. That’s equal to about 7.5 million fewer people.

But a debate has raged over how many of them are waiting for the economy to strengthen further before they look.

At least half the exodus is due to retirements by the vast baby boom generation. Younger adults are now also likely to stay in school. And some jobless people who aren’t seeking work are now receiving disability aid.

To some economists, all this means most of the dropouts aren’t coming back. If so, employers will soon have to pay more because the pool of potential workers has shrunk.

Others, like David Blanchflower of Dartmouth College, think employers still regard many of the dropouts as potential hires and, partly for that reason, feel scant pressure to raise pay. Andrew Levin, an economist at the International Monetary Fund, points to surveys showing that many of those not looking for work would return for the right job.

At a news conference last month, Fed Chair Janet Yellen said she thought a “meaningful” number of the dropouts would take jobs if more were available.

— What’s happened to 3.8 million people who had been unemployed for over six months but no longer are?

The decline in long-term unemployed is encouraging. But here’s what we don’t know: How many of them have actually gotten jobs? And how many have just given up looking and so are no longer counted as unemployed?

Labor Department figures show that no more than about 12 percent of the long-term unemployed are finding jobs in any given month — below pre-recession levels of about 18 percent.

But two Fed economists who studied changes over a full year found a brighter picture: Over a 12-month period, nearly 40 percent of the long-term unemployed find jobs. Only about 32 percent drop out.

If many of the long-term unemployed are getting jobs again, it suggests that the Fed might keep rates ultra-low longer to try to fuel further hiring. On the other hand, if most aren’t being hired, it might show that even a strengthening economy with low rates is no longer helping much.

— What does it mean that so many people — 7.3 million — who want full-time jobs can find only part-time work?

Before the recession, this figure was just 4.6 million. Levin, the IMF economist, thinks it shows there are lots more people who want additional work than the unemployment rate suggests. He estimates that the ranks of so-called involuntary part-timers are equivalent to an additional 1 point in the unemployment rate. As a result, Levin argues, paychecks aren’t likely to rise anytime soon.

Some disagree. Richard Fisher, head of the Federal Reserve Bank of Dallas, said last month that wages typically start rising once unemployment falls to the current 6.1 percent. He warned that wages may do so again, igniting faster inflation.

Research by the Federal Reserve Bank of Chicago indicates that many of the long-term unemployed, when they do find jobs, take part-time work even though they want full-time jobs. This trend could keep the number of involuntary part-time workers elevated.

— Companies are advertising lots more jobs. So why aren’t they filling more of them?

The number of available jobs has fully recovered from the recession. Yet total hiring hasn’t.

An index compiled by Steven Davis, a University of Chicago economist, indicates that companies aren’t trying to fill jobs as fast they were before the recession. Companies took an average of nearly 25 days to fill a job in July 2014, up from fewer than 22 in 2006.

There are several possible reasons. Companies may be choosier than they were before the recession because they feel that with unemployment still relatively elevated, they can be picky.

Michael Hanson, an economist at Bank of America Merrill Lynch, also notes that technology has made it so easy and cheap to post jobs that companies may do so even when they’re not ready to hire. They may just want to see their options.

“It’s one big fishing expedition,” Hanson says.

— Why has job growth fallen for women?

U.S. employers have added an average of 215,000 jobs a month this year, up from 194,000 in 2013. But job growth for women has dropped more than 14 percent this year.

Since the recession officially ended in 2009, women have taken fewer than 40 percent of the jobs added. At the same point in the previous three recoveries, they’d accounted for much bigger shares: 47 percent after the 2001 recession, 51 percent after the 1990-1991 recession and nearly 63 percent after the 1981-1982 recession.

A government report has found that men have been gaining jobs in businesses traditionally dominated by women, such as hotels, restaurants and financial services. Women have also lost ground in manufacturing.

Heidi Hartmann of the Institute for Women’s Policy Research notes that women tend to work in factories that make textiles and other non-durable goods. These jobs have been especially vulnerable to foreign competition and layoffs.

Hartmann doubts the trend will continue. Men’s outsize job gains stem from the fact that they accounted for 71 percent of the jobs lost in the recession. Many of their jobs have bounced back, though they’re still 350,000 short of the number they had when the recession began. Once they’ve regained their lost jobs, hiring should even out.