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Kill bill: How Illinois' temp industry lobbying quashed reform
By Will Evans / June 13, 2016
Supporters of Illinois Senate Bill 47 said it would provide proof that temp agencies were shutting out black workers in favor of Latinos they could more easily exploit. But the legislation never got a final vote.
Credit: deepak/flickr.com
The code word for black workers caught Rosa Ceja by surprise.
She was working for a temp agency in spring 2014, supervising workers at a big brick packaging plant northwest of Chicago. Crews of minimum-wage temp workers in hairnets boxed consumer products such as adult diapers and energy drinks in shifts around the clock.
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She knew the company wanted only men for some jobs and only women for others. And she knew those codes: "heavies" for men, "lights" for women.
But when Ceja asked the recruiting office to send more heavies, she was told there were only "guapos" available. She was confused. "Guapo" means "good-looking" in Spanish. "I'm like, 'Who cares if he's cute?' " Ceja remembers saying.
Guapo, her fellow recruiters told her, meant a black worker. Black people didn't want to work hard or get their hands dirty, they explained, so they were called the pretty ones. Latinos, the "feos" or ugly ones, were what the company wanted.
Ceja said she checked with a packing company supervisor to make sure.
"She said, 'If you ever bring me a black guy, they're automatically walking out,' " Ceja said.
Ceja had stumbled into the noxious muck of systemic discrimination in the rapidly growing temp industry. Around the country, temp agencies have used code words, symbols and gestures to illegally hire workers by sex, race and age. Whether filling orders for only white workers or only Latinos ones, the tactics often hit black workers the hardest.
As Ceja underwent a crash course in this hidden system, an advocacy group called the Chicago Workers' Collaborative was devising a counteroffensive.
The group eventually developed legislation to make temp agencies keep track of the race and gender of all job applicants.
Illinois Senate Bill 47 would provide the proof, supporters said, that temp agencies were shutting out black workers in favor of Latinos they could more easily exploit. If it passed, it could mean a fair chance at a paycheck.
Although a modest reform, it was likely the first of its kind. It put Illinois at the forefront of tackling a national problem in an increasingly important part of the workforce.
This is the story of why SB 47 was born – and how it was killed.
Rosa Ceja, a former dispatcher for Most Valuable Personnel, said the temp agency used the code words "guapos" and "feos" to refer to black and Latino workers, respectively. She was told not to send guapos.Credit: Univision
Inside Illinois' silvery-domed Capitol building in May 2015, Democratic Rep. Ken Dunkin was trying to guide the bill through a Labor and Commerce Committee hearing. It already had cruised through the state Senate the month before.
But a powerful Latino lawmaker pushed back, and not gently. Staffing agencies were doing a great job employing his mostly Latino constituents, he said. The exchange got heated.
"You're hurting people in my community," said Rep. Luis Arroyo, also a Democrat and a power player on Chicago's Northwest Side. "You're hurting the people that I'm putting to work in my community."
Dunkin tried to say the bill, which would amend the Day and Temporary Labor Services Act, wasn't just about boosting the job prospects of black workers. But Arroyo cut him off. If it wasn't just about black workers, Arroyo argued, why weren't any Latinos complaining about the situation?
"I don't want folks to get this twisted," Dunkin said.
"It's twisted already," Arroyo shot back.
Arroyo's district is 61 percent Latino and 4 percent black. The Chicago district Dunkin held – and lost in the March primary – is just about the opposite, at 52 percent black and 4 percent Latino.
The debate echoed the tensions between black and Latino laborers competing for minimum-wage jobs on the streets of Chicago.
It's a dynamic playing out in cities across the country, as some employers choose Latinos for jobs black workers once often had. The clash is particularly sharp in Illinois, with one of the highest rates of black unemployment, far surpassing Latino joblessness.
And that wasn't the only problem.
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Dan Shomon, a lobbyist for an alliance of temp agencies, said the proposed law would create a paperwork nightmare costing businesses millions of dollars. Shomon is also executive director of the Staffing Services Association of Illinois, which he said represents 25 agencies that provide 250,000 jobs a year.
"We oppose discrimination also," said Shomon, a top aide to Barack Obama during his state and U.S. Senate careers until 2006. He didn't mention that, just one week before, a board member of his association had signed an $800,000 settlement to resolve government findings of widespread discrimination at his company. Other association members also have been hit with bias claims.
Shomon said the bill's requirement to track the race and gender of job applicants wasn't necessary: The federal government already collects that data, he told legislators. He had that wrong, but nobody caught the error. He repeated it a few times.
After the SB 47 hearing, Shomon took Arroyo and another bill opponent to Saputo's, an Italian restaurant popular with politicos, spending $45 on each of them, according to lobbyist expenditure records. Asked about this, Arroyo said free meals don't buy votes in Springfield.
"I would be a heavy, heavy individual if that was the case," he said.
Following another hearing on the bill a few days later, the lobbyist treated Arroyo to a $50 meal at Nick & Nino's, a "penthouse steakhouse" with 30th-floor views of the Capitol dome. By then, the bill effectively was dead.
At the second hearing, Arroyo and another lawmaker had made Dunkin promise that he wouldn't bring it up for a final vote unless the industry lobbyist approved it.
And with that, a lobbyist who spread inaccurate information, representing companies that have faced serious allegations of discrimination, squelched what could have been a trailblazing reform.
SB 47 had grown out of a larger battle over hiring discrimination in the temp industry.
The Chicago Workers' Collaborative has been raising a ruckus for years as one of the nation's only labor groups running a grassroots campaign to reform temp agency hiring practices. The regional office of the federal Equal Employment Opportunity Commission is among the most active in the country in suing over temp industry job bias. And a Chicago union organizer-turned-workers rights attorney has unleashed multiple class-action lawsuits against temp agencies on behalf of black workers, recently clinching a $1.5 million settlement.
The temp agency tactics aren't lost on job seekers who show up at stark storefront recruiting offices before dawn. Some take buses across Chicago as early as 4 a.m. – not for a job they can count on, but for a chance to snag a shift making as little as $8.25 per hour.
Former temp worker Derell Pruitt said that at one job site, a packaging plant, a supervisor picked all the Latinos and sent back all the black workers. "It was very bold and humiliating, too," he said.Credit: Will Evans/Reveal
They sit next to each other with packed lunches, in rows of plastic chairs or wooden benches. They wait for hours to see whether they will get a spot on the vans that transport temp workers to various plastics manufacturers, food factories, printing plants and packaging facilities.
For Derell Pruitt, it was bad enough watching Latino workers board the vans while black workers like him were left behind. The worst, he said, was one time last year when he actually did get sent out for a job. It was for a temp agency called Most Valuable Personnel, or MVP.
As soon as the vanload of workers arrived at the job site, a packaging plant, Pruitt said a supervisor picked all the Latinos and sent back all the black workers. They weren't needed, they were told.
"I could not believe that they would not pick one black, not one," said Pruitt, talking in a McDonald's near his Waukegan home. "It was very bold and humiliating, too."
Rosa Ceja said the same thing happened on her watch. Back when she learned that black workers were "guapos" to be avoided, she was a dispatcher for MVP, working on-site at the same packaging plant.
"Right now, if I think about it, it's really, really sad," she said.
Another former dispatcher, Adriana Hernandez of Round Lake, has similar stories: "They would tell me, 'Remember, the supervisor doesn't want any older ladies, and she doesn't want any black people.' " She provided a text message that she claimed showed the codes for men and women, an order for "10 lights and 4 heavys."
In response to the allegations, an attorney for MVP, Elliot Richardson, said, "We deny that we discriminate on any basis, including race or gender or for any other reasons whatsoever."
He declined to address specifics.
"It's not our desire to be litigating this in the media," Richardson said. "We have directed our clients not to speak to the media about these pending matters."
The packaging plant, called Metropolitan Graphic Arts, or MGA, doesn't hire by race, said its president, Greg Szymanski. If any workers were sent back, he said, it was because they didn't want to do the job.
"We'll hire the United Nations, and we'll look for good workers, period," he said.
But Szymanski acknowledged requesting male workers for certain jobs and women for others.
"If it's picking up very heavy things, men are stronger, sure," he said. "We've been in this business, and we know that men do this job better and women do this job better."
A text message sent to former MVP dispatcher Adriana Hernandez contains what she says were the codes for men and women, an order for "10 lights and 4 heavys." Credit: Courtesy of Adriana Hernandez
MGA's clients include a long list of name-brand companies that once scrolled across its website: Advil, Dove, Crest, Tide, Huggies, Tums, L'Oreal, Arm & Hammer, Target, Starbucks, Walmart and Unilever.
Pruitt is part of class-action lawsuits claiming that MVP discriminated against black workers and didn't pay him for his time. MVP responded in court filings that Pruitt never worked for the company.
Pruitt, 40, said he had to go to counseling for his rage – against Latinos.
"I was starting to hate them," he said. "I knew that I shouldn't be thinking the way that I was thinking, but I was frustrated and angry."
Pruitt said he has gone from backing Obama to full-throated support for Donald Trump, citing the Republican presidential candidate's combative stance on immigration.
"I want to help build that wall," he said of Trump's plan to seal off the U.S. border with Mexico. "I want to help get rid of some of these people."
Racial preferences loom over recruiting offices around the country, lawsuits and interviews show.
A trio of temp agencies in Tennessee, for example, systematically recruited Latinos at the expense of black workers, according to government lawsuits.
"Whether it's low-wage Hispanic workers or low-wage African American workers, each of them are trying to make a decent living, and they're being pitted against each other," said Faye Williams, regional attorney for the Memphis office of the Equal Employment Opportunity Commission.
This dynamic goes back to the 1990s, when global economic competition drove employers to seek a cheaper, more flexible workforce, said Virginia Tech sociology professor Barbara Ellen Smith.
They found a solution in temp agencies, which allowed for quick and easy hiring and firing. And they found a willing workforce in a growing stream of Latino immigrants, desperate for work of any kind, she said.
"In effect, employers were replacing black workers with Latino immigrant workers," said Smith, who studied the phenomenon among warehouse workers in Memphis. "You've got a population that's really kind of exploitable."
The ensuing conflict between black and Latino workers served employers' interests, too: "It keeps people from organizing," she said.
Overcoming that racial divide has been the ongoing struggle of the Chicago Workers' Collaborative. The group, formed in 2000, traditionally advocated for Latino workers, campaigning against problems such as wage theft and sexual harassment. But a few years ago, its leaders reached out to black workers and discovered a reservoir of frustration and resentment.
"Their view was those 10 Latinos … getting in that van are the reason I'm not going to work," said Leone Bicchieri, the group's executive director. "As opposed to, it's the dispatcher following orders from someone higher up."
Black workers, he said, are more likely to complain about wage theft or workplace injuries than Latino workers, who might be worried about their immigration status or unsure of other options.
Bicchieri's wife emigrated from Mexico and worked for years as a temp. He's trying to convince Latino workers it's in their long-term interests to fight job discrimination that actually favors them.
The group's pitch is this: Black workers are losing jobs, and Latinos are losing their rights. Only united can they hold temp agencies accountable.
Leone Bicchieri is executive director of the Chicago Workers' Collaborative, which traditionally advocated for Latino temp workers. But a few years ago, he says the group reached out to black workers and discovered a reservoir of frustration and resentment.Credit: Stacey Rupolo/The Chicago Reporter
Rejected black workers left sitting in temp agency lobbies may feel discrimination, but they can't quantify it.
Despite lobbyist Dan Shomon's repeated assurances that federal regulators already collect the data, it's just not true. Officials with both the American Staffing Association and the Equal Employment Opportunity Commission agree on that point.
The federal government compiles the demographics only of employees, not of job applicants. And the temp industry pushed for a federal exemption to those reporting requirements in the 1960s.
There is a state law in Illinois requiring agencies to keep a record of the race and gender of each temp worker who is hired. But it appears that law isn't always followed or enforced.
MVP lawyers told a judge in 2014 that the company doesn't record the race of its workers and doesn't keep any record of job applicants.
The Illinois Department of Labor doesn't inspect temp agency records unless it receives a complaint, and there haven't been any related to race and gender recordkeeping, according to the department.
That raises the question: Even if the reform were enacted, would new rules also be ignored?
Shomon, the lobbyist, declined to answer questions from Reveal or provide information that backed up his assertions to lawmakers. Instead, he pointed to the long-standing federal law against hiring bias, writing by email, "This discrimination is already prohibited by the EEOC."
The nonprofit trade group he represents came together, according to its website, to "fight to keep our industry free of unnecessary regulations."
Its leadership is quite familiar with the discrimination problem.
The secretary of the board of directors is Chuck Porcelli, CEO of Bensenville-based ASG Staffing, recent federal and state filings show. A former office manager sued the company in 2014, accusing it of catering to customer requests for workers of a certain race or sex. The company used a coding system, the suit said, for "racial 'earmarking.' " A "Code 3, heavy," for example, meant a request for a Latino man.
The company denied the allegations and settled confidentially with the former employee. ASG's attorney, Bethany Drucker, said it does not discriminate.
Another association board member, Scott Reedy, is president of the Chicago-area firm Source One Staffing. His company agreed to pay $800,000 last year to resolve two lawsuits by the Equal Employment Opportunity Commission. The government accused Source One of discriminating against temp workers by race, gender and disability.
"That's a significant case and significant relief with respect to serious systemic discrimination," said commission regional attorney John Hendrickson.
It all started in 2007, when a few female temp workers complained that a supervisor at a West Chicago plastics plant had sexually harassed them.
The government found that Source One segregated jobs into "men's work" and "women's work" and sent women to inferior, lower-paying jobs. The company sometimes assigned women to hostile work environments and then retaliated against those who complained.
Illinois Rep. Luis Arroyo, D-Chicago, fought SB 47 partly because he doesn't believe there's racial discrimination in temp worker hiring. "They tried to say that the temp agencies only hire Latinos," he said. "I think that that's wrong to say that. … Do I know that for a fact? No, I don't." Credit: Bryan Bedell/Galewood.net
One client, according to email requests, repeatedly asked for "girls." Another, according to a deposition, wanted only "pretty girls."
As for race, Source One also filled client requests for Latino workers, discriminating against white and black applicants, according to the commission.
Reedy, the company president and association board member, was personally swept up into the case. A former employee testified that he instructed her to fill gender-specific requests, but to lie to government investigators about what was happening.
The government called it witness tampering and said it "raises serious questions about unethical or even criminal conduct by Source One's owner and/or its attorney."
Source One denied wrongdoing but agreed to the settlement and ongoing supervision by an independent monitor. Reedy had to personally guarantee $150,000 of the settlement, a requirement Hendrickson, the Equal Employment Opportunity Commission attorney, called "extraordinarily rare."
Reedy, reached by phone, said of the government allegations: "That's done and gone, and I already spent eight years on it." He declined to answer other questions, saying, "I don't have any comments regarding anything," and hung up the phone.
Companies run by other association board members face pending discrimination complaints with the commission and in court.
At the Springfield hearings for SB 47, though, legislators seemed to doubt whether there was a problem to fix at all. Rep. Luis Arroyo, for one, didn't believe it.
"They tried to say that the temp agencies only hire Latinos. I think that that's wrong to say that," Arroyo said in a January interview. "Did I do any investigations? Do I know that for a fact? No, I don't."
Pamela Sanchez tried to tell them. She worked as a Most Valuable Personnel dispatcher in Cicero, a Chicago suburb, until February 2015. The workers collaborative invited her to Springfield to testify at a committee hearing in May 2015. She brought her daughter.
Rep. Elizabeth "Lisa" Hernandez, who represents Cicero, asked Sanchez whether temp workers are chosen on a first-come, first-served basis. She was surprised at the answer.
"No, ma'am," Sanchez replied. "Most of the time, people will choose by race."
Some wrote off the dispatcher's firsthand account. The rumor going around, Arroyo said, was that Sanchez had been paid to say those things.
"That's what they were saying in the halls and in the back," he said.
Pamela Sanchez was a dispatcher for Most Valuable Personnel in Cicero, Ill., until February 2015. She testified before a state legislative committee that many temp workers are selected by race.Credit: Stacey Rupolo/The Chicago Reporter
After all, Arroyo said, "Why would a Latina say that they are only hiring Latinos?" It didn't make sense that she would speak out against something that benefited Latinos.
"Latinos normally don't do that," he said.
Sanchez, a soft-spoken single mother of two from Chicago's West Lawn neighborhood, said she went public because she used to be a temp worker herself.
"I know how hard it is when you need to be there and looking for a job and they don't pick you," she said. "I cannot be shutting my mouth no more."
There was also an element of guilt, of making things right. She says she still feels bad about the time she confirmed a black worker for a temp job, telling him to come into the office the following day around 4 a.m. He waited for five hours, she said, but was passed over. She said her manager told her, "Oh, you know they don't want no blacks."
Sanchez said MVP fired her last year after she complained to upper management about problems at the Cicero office. Shortly afterward, she signed a court declaration for lawyers suing MVP over discrimination. It makes it clear that Sanchez had much more to say, had legislators asked.
In it, she describes the multitude of ways the system is rigged against black workers:
MVP had her pass out fliers advertising work in Spanish only, she said. Van drivers picked up laborers in Latino neighborhoods such as Little Village. Recruiters like herself kept their own lists of Latino workers to call if a job came open.
Latinos who ventured into a recruiting office were given an application and often sent out to work right away. Black job seekers were told they couldn't fill out an application unless there was a job opening. They had to sign an authorization to run a criminal background check, but Latino workers didn't.
MVP, Sanchez said, would send black workers to jobs no one else wanted, such as meatpacking plants. The cold conditions were so uncomfortable that Latino workers asked to be transferred elsewhere. With no other option, the black workers would go, she said.
Elite Staffing is a founding member of the Staffing Services Association of Illinois, which lobbied against state legislation to make temp agencies keep track of the race and gender of all job applicants.Credit: Stacey Rupolo/The Chicago Reporter
MVP recently transferred its Illinois branches, including the Cicero one, to Elite Staffing, but maintains branches in other states. Elite Staffing is a founding member of the Staffing Services Association of Illinois, and an Elite executive currently serves as board president.
As for Arroyo's rumor about her testimony, Sanchez said she wasn't paid a penny. Arroyo, however, was.
Three months after the hearing at which Arroyo lambasted SB 47's sponsor for "hurting my community," several temp agencies gathered for a fundraising dinner Aug. 24. Six agencies pooled together $5,000 for Arroyo's campaign account, according to campaign finance records.
One of the donors was Personnel Staffing Group, the parent company of MVP, where Sanchez worked.
Another was ASG Staffing, the company accused of coding workers by race and sex.
A third was ClearStaff, a Woodridge-based agency sued in 2014 by a former sales representative who complained about office talk involving racial slurs and sexual favors for clients. She lost her suit but had prompted an Equal Employment Opportunity Commission investigation.
The commission found reasonable cause that the company discriminated against black job applicants and paid Latino workers less than non-Latinos. The commission couldn't come to a resolution with ClearStaff and decided not to file suit, according to court records. ClearStaff's president, Richard Seeman, formerly served on the board of the industry association. He didn't respond to phone calls.
This wasn't a routine show of support for Arroyo. None of the agencies had donated to him before, campaign contribution records show. But Arroyo said the donations had nothing to do with his stance on SB 47.
"I've always supported the temp agencies, because it works in the Northwest Side, he said. "You got to support the people that support you."
Chris Williams of the Workers' Law Office in Chicago has sued six temp agencies and many of their clients since 2012 for excluding black applicants. Credit: Stacey Rupolo/The Chicago Reporter
Temp agencies opposed the bill in part because they worried it would fuel discrimination lawsuits against them.
Indeed, data on the disparity between applicants and hires can form crucial evidence in such cases. And Chicago-area agencies already are facing a few.
Chris Williams has become the attorney to fear. Since 2012, his Workers' Law Office has sued six temp agencies and many of their clients for excluding black applicants. That's on top of many other wage-related lawsuits and discrimination complaints filed with federal regulators.
Williams went from teaching English learners for the City Colleges of Chicago to union organizing in the 1990s. In 2000, he helped launch what became the Chicago Workers' Collaborative.
After he was admitted to the Illinois bar in 2004, Williams co-founded and led a legal clinic to help immigrant workers who had been cheated on wages. That's where he first heard about discrimination against black workers at temp agencies.
It all fit together, Williams said.
"There is a desire to hire workers who are more vulnerable and thus subject to exploitative working conditions … which then reduces labor costs," he said.
Even permanent workers aren't immune, he said. "It drives down wages for everybody."
Temp agencies, he said, are worried that his legal onslaught and the reform legislation is an "attack on their business model." In December, the candy manufacturer that makes Lemonheads, Red Hots and Jujyfruits, plus two of its temp agency suppliers, agreed to a preliminary settlement of $1.5 million in a racial discrimination lawsuit.
"He is notorious for bringing these cases," said Scott Polen, whose company, TempsNow, was sued last month.
Polen, a board member of the temp industry association, said Williams brings unfounded lawsuits to milk agencies for settlement money. TempsNow, he says, doesn't take race into account.
"Quite honestly, because of Chris Williams, I wouldn't even contemplate doing something like that," he said, "because it's just not worth it, not to mention it's wrong."
Another lawsuit, filed last year, focuses on the workers Most Valuable Personnel supplied to Gold Standard Baking. The mammoth brick-walled industrial bakery near railroad tracks on the Southwest Side churns out more than 10 million croissants a week for convenience stores and supermarkets around the country.
Of some 5,000 temps provided to the bakery by MVP over several years, 98 percent had Latino surnames, according to the suit. MVP disputes those numbers.
In court filings, company lawyers called the suit "nothing more than a charade, designed to harass Defendants." The plaintiffs "are throwing metaphoric noodles at the wall, hoping that any allegation will stick."
Beyond the legal sparring, though, there are the accounts of insiders such as former bakery supervisor Maria Carretero. Bakery managers, she said in a court declaration, called black workers lazy and worthless and told MVP not to send them.
"Quitan esos negros," a plant manager yelled angrily in Spanish, she said. "Get rid of those blacks."
"It was common knowledge that GSB (Gold Standard Baking) did not want to have African Americans at the factory," Carretero's statement said.
Reached by phone, a bakery executive, Mark Grosshans, said, "I don't want to discuss this."
Pamela Sanchez, the former MVP dispatcher who testified at the Springfield hearing, worked on-site at the bakery for several months. Black workers, she said, usually were assigned only on weekends to avoid paying Latino workers overtime or to graveyard shifts that were hard to fill. Even then, Sanchez said the bakery told her to "DNR" ("do not return") those black workers after their shift.
Job orders became hard to fill as Latino workers complained about the heat, the speed of the machines and the lack of breaks, she said. But the bakery still didn't want black workers.
Sanchez recently sued MVP and the bakery for retaliating against her after she complained about discrimination.
An MVP lawyer said in court that the reason for its largely Latino workforce isn't nefarious at all: The agency's Cicero office is in a predominantly Latino neighborhood.
Many temp agencies are. A 2001 academic paper found that Chicago temp agencies strategically placed their recruiting offices in Latino neighborhoods to attract the kind of workforce their clients wanted.
Black workers, of course, can travel. Some trek long distances. And as Williams notes, just across the street from that Cicero office is a mostly black neighborhood.
Without hard data, the other way to ferret out discrimination is through whistleblowers. But many are afraid to stick out their necks – and with good reason.
As SB 47 stumbled through the Illinois General Assembly last year, Rosa Ceja faced her own crucible. The former MVP dispatcher had spoken out, and the company sued her.
It was for something she wrote on Facebook.
Ceja's Facebook feed often features photos of herself in heavy makeup, sometimes mugging for the camera with pursed lips – the kind of beauty-shot selfies that abound on social media. Other times, her posts feature the young niece and nephew she raises in her Waukegan-area house.
But one day, Ceja wrote, in Spanish, "If MVP (Most Valuable Personnel) owes you hours (or) you've been victims of abuse please send me a message there are people interested in ending the injustices they are doing. Please share."
Ceja was forced out of the company in 2014, she said, after complaining about sexual harassment by an MVP van driver. He recently was convicted of battery against her. MVP contends that Ceja abandoned her position.
Afterward, she, too, signed a statement about the discrimination she witnessed. She accused top managers of being complicit in rejecting black workers. The company, in its legal response, called it hearsay.
When Ceja posted on Facebook in December 2014 to see whether anyone else had problems with MVP, she wasn't just poking around for fun. Federal investigators had asked her if she could find other victims or witnesses, according to her court filings.
Her Facebook friends asked her what she was talking about. Ceja replied: "People sometimes don't get paid or if they report that they have been sexually abused they don't care."
The day before SB 47 was introduced in January 2015, MVP sued Ceja for defamation. "Ceja's statements have damaged MVP's reputation in the industry and the community," the suit said.
The company demanded at least $50,000.
Ceja panicked. She thought she'd lose her house. She stopped answering her phone. She couldn't sleep. The company sent subpoenas to her Facebook friends who had commented on her post. Some were furious, Ceja said, that she had pulled them into a legal dispute.
The lawsuit dragged on as legislators argued over SB 47. Then last summer, the judge dismissed the suit. Ceja's post was opinion, protected by the First Amendment.
Separately, the Equal Employment Opportunity Commission found reasonable cause that MVP's lawsuit constituted illegal retaliation against her. The National Labor Relations Board issued a complaint that the company's lawsuit violated Ceja's rights.
But it had taken a toll. Ceja says she lost friends and job opportunities. And she's still nervous. She doesn't want anyone to know her address.
The lawsuit also has had a broader chilling effect, said Chris Williams, Ceja's lawyer.
"People are very aware that Rosa has been sued," he said, "and as a result are very nervous about speaking publicly."
Illinois Rep. Carol Ammons, D-Champaign, wants to revive legislation targeting discrimination in temp worker hiring. "The industry is exploiting Latino workers, black workers, white workers," she said. "The goal now is to expose the exploiter."Credit: Seth Perlman/Associated Press
At the final May hearing for SB 47 last year, Rep. Luis Arroyo said he had met with Rep. Ken Dunkin, the bill's sponsor, and Dan Shomon, the industry lobbyist, to work out an agreement. Arroyo would vote the bill out of committee, but only if Dunkin agreed not to bring it to a final House vote without the opposition's approval.
Dunkin tried to give himself some wiggle room. He would try to find a compromise, he said, but questioned the opposition's willingness to negotiate sincerely.
"There's been a whole bunch of unnecessary games being played with this, and I'm right in the middle," he said at the hearing.
Republican Rep. Mike Tryon nailed him down. Tryon was named legislator of the year by Shomon's alliance of temp agencies in 2011. An announcement at the time lauded him for doing "his best to make sure their needs are met."
Tryon made Dunkin commit publicly that he wouldn't push the bill further without the industry's blessing. No compromise ever was reached, and the bill never got a final vote. Dunkin, who wouldn't give an interview, lost to a primary challenger this year.
Now, Democratic Rep. Carol Ammons, whose career in Springfield is only as old as SB 47, wants to revive the fight.
Winning her seat in an upset to become the first black representative of the Champaign-Urbana district, Ammons is ambitious. She doesn't think SB 47 went far enough.
"The industry is exploiting Latino workers, black workers, white workers," she said. "The goal now is to expose the exploiter."
Arroyo, for his part, said he'd back a bill like SB 47 if he had any evidence of discrimination.
"If there is an investigation and there is a fact that they only hire Latinos, I wouldn't be for that," he said. "I will get on the bill right away, and I would support the bill."
This story was edited by Fernando Diaz and Amy Pyle and copy edited by Sheela Kamath and Nikki Frick.
Will Evans can be reached at wevans@revealnews.org. Follow him on Twitter: @willCIR.
By Will Evans Reveal from The Center for Investigative Reporting June 13, 2016
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MVP had her pass out fliers advertising work in Spanish only, she said. Van drivers picked up laborers in Latino neighborhoods such as Little Village. Recruiters like herself kept their own lists of Latino workers to call if a job came open. Latinos who ventured into a recruiting office were given an application and often sent out to work right away. Black job seekers were told they couldn't fill out an application unless there was a job opening. They had to sign an authorization to run a criminal background check, but Latino workers didn't. MVP, Sanchez said, would send black workers to jobs no one else wanted, such as meatpacking plants. The cold conditions were so uncomfortable that Latino workers asked to be transferred elsewhere. With no other option, the black workers would go, she said. Elite Staffing is a founding member of the Staffing Services Association of Illinois, which lobbied against state legislation to make temp agencies keep track of the race and gender of all job applicants.Credit: Stacey Rupolo/The Chicago Reporter


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I feel like I made this process SO MUCH HARDER than it really needed to be by over-thinking everything, but in the end I settled for the above nine. A beardy boyfriend (mostly a tribute to his beard, which got shaved immediately after our holiday, sob!), lovely places from holidays here and abroad, a little memento to Max, bits and pieces from my flat, favourite spots in Edinburgh, and my pussycat.
They arrived super quickly, and I then realised that the only magnetic spot in my house was my bedroom radiator... but luckily, I think they work quite well in that space!
I've not actually sorted out my books since I first moved into my flat, and am now so used to my messy stacks of books that I think it's super unlikely I'll ever get round to arranging them neatly.
Overall, I'm really pleased with the magnets! They're of super good quality, and I think they'd make a really nice gift (maybe check the recipient has a fridge first!). You can also get a jigsaw version of one photo, that I think would be really fun for kids.
If you're interested in trying some yourself, then you can get 15% off with the code FRIEND0PBL (just rolls off the tongue, doesn't it).
Can you pinch other people's photos, or do they have to be one's (slightly rubbish) own? Asking for a friend.
Web based life marketing by means of Instagram can encourage you accomplish your focused on business objectives no sweat. Use Instagram widely and fittingly, and become greater!


We specialize in buying and selling homes throughout Macon, and WARNER ROBINS! If you are a buyer or seller who lives near Maconor WARNER ROBINS, or if you are looking to relocate to our area, we are happy to assist you! Real estate is one of the largest and most exciting investments one can make. We are here to make sure that your process is a fun and rewarding one! Our website is geared toward our local areas of Macon and WARNER ROBINS, and also surrounding communities such as Byron, Gray, Forsyth, and Atlanta. We are specialists in our community and we would be happy to help you with all of your real estate needs!
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The second of our new weekly walking tours for small groups of between 7 and 10 participants, this one will explore the historic coast-line of Chinatown to explore the traditions, customs and religions brought here by large numbers of merchants, traders and coolies.
You will start on the south bank of the Singapore River and walk along Chulia Street, the former enclave of South Indian Chettiars. On the way to Telok Ayer Street you will learn more about the great diversity of the Chinese community in mid-19th century Singapore, and the spirit of mutual assistance as provided by the Clan Associations.
You will visit UNESCO awarded Wak Hai Cheng Bio and Tien Hock Keng temples – great examples of SEA temple architecture - where you will meet gods and goddesses and learn about Chinese religions and symbolism.
You will discover that Chinatown was not only a Chinese enclave by visiting one of the oldest Indian Muslim mosques. While admiring the beautiful facades of shophouses, you will climb to the highest point of Chinatown, Ann Siang Hill, before rounding our tour off at the URA (Urban Redevelopment Authority) with its very informative City Gallery.
These walking tours will have a maximum of 10 participants but also need a minimum number of 6 to go ahead; if we don't have enough people we will inform those who have registered (& refund their payment) by the evening before at the very latest. But we hope they will always run!!
"I would like to thank you for the tour yesterday, it's been a wonderful morning! Your guide is amazing and so expert about history".
"I just wanted to let you know that my aunt had a brilliant time yesterday. She couldn't say enough about the tour and how great your guide was. Thanks so much! I will certainly recommend you".
"Your guide was WONDERFUL! We all enjoyed her. I'm looking forward to the next tour."


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text:
\section{Introduction}

In certain situations, geometrical information of manifolds might be
encoded in differential forms. In the presence of symmetries of the
manifold via the action of a Lie group, the behavior of these
differential forms under the group action may lead to a better
understanding of the manifold itself. In some well known instances
of actions of compact Lie groups, the action is of a particular type
whenever the differential form may be extended to an equivariant one
in the Cartan model of equivariant cohomology \cite{Cartan}; this is
the case for example in Hamiltonian actions on symplectic manifolds
\cite{AtiyahBott}, in Hamiltonian actions on exact Courant
algebroids \cite{Bursztyn, CaviedesHuUribe, Hu2, Uribe-dg} or in
gauged WZW actions which are anomaly free \cite{Witten} and its
generalizations \cite{GarciaCompeanPaniagua, Paniagua}.

When the Lie group is not compact, the Cartan complex associated to
the action in general does not compute the equivariant cohomology of
the manifold, and therefore many of the results that hold for
compact Lie groups may not hold for the non-compact case. But since
the Cartan model is very well suited for studying the infinitesimal
behavior of the differential forms with respect to the action of the
Lie algebra, it would be worthwhile knowing more about the relation
between the cohomology of the Cartan complex and the equivariant
cohomology. In this paper we study this relation and we obtain some
interesting results which in particular permit us generalize the
conditions for cancellation of anomalies on gauge WZW actions
developed by Witten \cite{Witten}, to the case on which the gauge
group is the non-compact group $SL(n,\ensuremath{{\mathbb R}})$.

The main ingredient of this work is the equivariant De Rham complex
developed by Getzler \cite{Getzler} whose cohomology calculates the
equivariant cohomology independent whether the group is compact or
not. We show that there is an inclusion of complexes of the Cartan
complex into the equivariant De Rham one, and therefore we obtain a
homomorphism between the cohomology of the Cartan complex to the
equivariant De Rham cohomology. This in particular implies that any
closed form in the Cartan complex defines an equivariant cohomology
class, but note that the converse may not be true. We furthermore
show that there is a spectral sequence converging to the equivariant
De Rham cohomology such that its $E_2^{*,0}$-term is isomorphic to
the cohomology of the Cartan complex, and in this way an explicit
relation between the cohomology of the Cartan complex and the
equivariant De Rham cohomology is obtained. Now, since the first
page of this spectral sequence could be understood in terms of the
differential cohomology of a Lie group, we recall its definition and
some of its properties and we reconstruct some calculations for the
group $SL(n,\ensuremath{{\mathbb R}})$.

We conclude this work by applying our results to the gauged WZW
actions whenever the gauge group is $SL(n,\ensuremath{{\mathbb R}})$. We recall the
result of Witten which claims that the gauged WZW action on compact
Lie groups is anomaly free, if and only if the WZW term $\omega =
\frac{1}{12 \pi} {\rm{Tr}}(g^{-1} dg)^3$ can be extended to a closed
form in the Cartan complex, and we generalize it to the case on
which the gauge group is $SL(n,\ensuremath{{\mathbb R}})$; this is Theorem
\ref{equivalence of extension for SL(n,R)}. We use this theorem to
construct explicit examples where the equations representing the
condition of anomalies cancelation hold and where they do not. The
physical implications of our work will appear elsewhere.

\vspace{0.5cm}

\noindent {\bf Acknowledgments} The first author was partly supported
 by the CONACyT-M\'exico through grant number 128761. The third author would
like to acknowledge and to thank the financial support provided by
the Alexander Von Humboldt Foundation and also would like to
 thank the hospitality of Prof. Wolfgang L\"uck at the Mathematics Institute of the
University of Bonn.

\section{Equivariant cohomology}

Let $G$ be a Lie group and $M$ a manifold on which $G$ acts on the
left by diffeomorphisms . The $G$-equivariant cohomology of $M$
could be defined as the singular cohomology of the homotopy quotient
$EG \times_G M$
$$H^*_G(M; \ensuremath{{\mathbb Z}}) : = H^*(EG \times_G M; \ensuremath{{\mathbb Z}}),$$
where $EG$ is the universal $G$-principal bundle $G \to EG \to
BG$.

The previous definition works for any topological group and any
continuous action, but sometimes it is convenient to have a De Rham
version with differentiable forms for the equivariant cohomology
whenever the group is of Lie type and the action is
differentiable.When the group $G$ is compact, the Weil and Cartan
models provide a framework on which the equivariant cohomology could
be obtained via a complex whose ingredients are the local action of
the Lie algebra $\mathfrak{g}$, and the differentiable forms
$\Omega^\bullet M$. When the group $G$ is not compact there is a
more elaborate model for equivariant cohomology that we will
describe in the next section.

\subsection{De Rham model of Equivariant cohomology} \label{equivariant De Rham complex}
One way to obtain a De Rham model for equivariant cohomology is through the total complex
of the double complex
$$\Omega^*\left(N_\bullet ( G \ltimes M)\right)$$
that is obtained after applying the differentiable forms functor to
the simplicial space $N_\bullet ( G \ltimes M)$ which is the nerve
of the differentiable groupoid $G \ltimes M$.

By the works of Bott-Shulman-Stasheff \cite{BottShulmanStasheff} and
Getzler \cite{Getzler} we know that one way to calculate the
cohomology of the total complex of the double complex of differentiable
forms
 $$\Omega^*\left(N_\bullet ( G \ltimes M)\right)$$
 is through the differentiable cohomology groups
 $$H^*(G,S{\mathfrak g}^* \otimes \Omega^\bullet M  )$$
of the group $G$ with values on the differentiable forms of $M$
tensor the symmetric algebra of the dual of the Lie algebra
$\mathfrak{g}$.

In \cite{Getzler} Getzler has shown that there is a De Rham theorem
for equivariant cohomology showing that there is an isomorphism of
rings
$$H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M ) \cong H^*(EG \times_G M ; \ensuremath{{\mathbb R}})$$
between the De Rham model for equivariant cohomology and the
cohomology of the homotopy quotient.

The De Rham model for equivariant cohomology defined by Getzler is
described as follows.

Consider the complex $C^k(G,  S{\mathfrak g}^* \otimes \Omega^\bullet M
)$ with elements smooth maps
$$f(g_1, \dots , g_k | X) : G^k \times \mathfrak{g} \to \Omega^\bullet M, $$
which vanish if any of the arguments $g_i$ equals the identity of
$G$. The operators $d$ and $\iota$ are defined by the formulas
\begin{eqnarray*}
 (df)(g_1, \dots , g_k | X) &=& (-1)^k df(g_1, \dots , g_k | X) \ \ \ \ \ \ {\rm{and}}\\
(\iota f) (g_1, \dots , g_k | X) &=& (-1)^k \iota(X) f(g_1, \dots
, g_k | X),
\end{eqnarray*} as in the case of the differential in Cartan's
model for equivariant cohomology \cite{Cartan, Guillemin}. Recall
that the elements in $\mathfrak{g}^*$ are defined to have degree 2, and
therefore the operator $\iota$ has degree 1. Denote the generators of the symmetric algebra
$S\mathfrak{g}^*$ by $\Omega^a$ where $a$ runs over a base of $\mathfrak{g}$.

The coboundary $\bar{d}: C^k \to C^{k+1}$ is defined by the formula
\begin{eqnarray*}
(\bar{d}f)(g_0, \dots , g_k|X) & = & f( g_1, \dots , g_k | X ) +
 \sum_{i=1}^k (-1)^i f(g_0, \dots, g_{i-1}g_i, \dots  , g_k | X)\\
 & & +(-1)^{k+1} g_k f(g_0, \dots , g_{k-1} | {\rm{Ad}}(g_k^{-1})X),
\end{eqnarray*}
and the contraction $\bar{\iota} : C^k \to C^{k-1}$ is defined by
the formula
\begin{eqnarray*}
(\bar{\iota}f)(g_1, \dots , g_{k-1}|X) & = & \sum_{i=0}^{k-1} (-1)^i
\frac{\partial}{\partial t}  f(g_1, \dots, g_i, e^{tX_i}, g_{i+1}
\dots  , g_{k-1} | X),
\end{eqnarray*}
where $X_i= {\rm{Ad}}(g_{i+1} \dots g_{k-1})X$.

If the image of the map
$$ f: G^k \to  S \mathfrak g^* \otimes \Omega^\bullet M $$
is a homogeneous polynomial of degree $l$, then the total degree
of the map $f$ equals $deg(f)=k+l$. It follows that the structural
maps $d, \iota, \bar{d}$ and $\bar{\iota}$ are degree 1 maps, and
the operator $$d_G = d + \iota +\bar{d} + \bar{\iota}$$ becomes a
degree 1 map that squares to zero.

\begin{definition}
The elements of the complex
$$\left( C^*(G,
S{\mathfrak g}^* \otimes \Omega^\bullet M ) , d_G \right)$$ will be called {\it equivariant De Rham forms} and
its cohomology $$H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M )$$
will be called the equivariant De Rham  cohomology.
\end{definition}

In \cite{Getzler} it was shown that the complex $\left( C^*(G,
S{\mathfrak g}^* \otimes \Omega^\bullet M ) , d_G \right)$ together with
the cup product
\begin{eqnarray*}
(a \cup b)(g_1, ... , g_{k+l}|X) = (-1)^{l(|a|-k)} \gamma
a(g_1,..., g_k| {\rm{Ad}}(\gamma^{ -1})X) b(g_{k+1},...,
g_{k+l}|X)
\end{eqnarray*}
for $\gamma = g_{k+1} ...g_{k+l}$, becomes a differential graded
algebra, and moreover that there is a canonical isomorphism of
rings
$$H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M ) \cong H^*(M \times_G
EG ; \ensuremath{{\mathbb R}})$$ with the cohomology of the homotopy quotient.

\subsubsection{Cartan model for equivariant cohomology} The Cartan model for equivariant cohomology is the differential graded algebra
$$C^*_G(M):=  ( S{\mathfrak g}^* \otimes \Omega^\bullet M)^G$$
endowed with the differential $d+\iota$. Therefore there is a natural homomorphism of
differential graded algebras \begin{eqnarray} \label{homomorphism Cartan to De Rham}
i: (C^*_G(M), d+\iota) \to \left( C^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M
) , d_G \right)\end{eqnarray} given by the inclusion
$$( S{\mathfrak g}^* \otimes \Omega^\bullet M)^G \subset C^0(G,
S{\mathfrak g}^* \otimes \Omega^\bullet M )$$ since the restriction of $d_G$ to
$( S{\mathfrak g}^* \otimes \Omega^\bullet M)^G$ is precisely $d + \iota$ because  the
operators $\bar{d}$ and $\bar{\iota}$ act trivially on the
invariant elements of $C^0(G, S{\mathfrak g}^* \otimes \Omega^\bullet
M)$.

The induced map on cohomologies
$$i : H^*(C^*_G(M), d+\iota) \to H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M
)$$ is far from being an isomorphism as the case of $M=pt$  and $G=GL(1,\ensuremath{{\mathbb R}})_+=\ensuremath{{\mathbb R}}_+^*$ shows.\\
 In this case $C_G(M)= S ( \mathfrak{gl}(1,\ensuremath{{\mathbb R}}))= \ensuremath{{\mathbb R}}[x]$ and $d+ \iota=0$, hence $H^*(C_G(M))=
\ensuremath{{\mathbb R}}[x]$ where $|x|=2$. On the other hand  $H^*(G,S{\mathfrak g}^*)= H^*(BG,\ensuremath{{\mathbb R}})=\ensuremath{{\mathbb R}}$ since $BG$ is contractible.

Nevertheless, when the Lie group $G$ is compact,
the map $i$ induces an isomorphism in cohomology \cite{Getzler}. Now, in order to understand in more detail the relation between the cohomology of the Cartan model and the equivariant cohomology we will introduce a spectral sequence suited for this purpose.

\subsubsection{A spectral sequence for the equivariant De Rham complex}
\label{subsubsection spectral sequence}

Let us filter the complex $C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M )$ by the degree in $S{\mathfrak g}^* \otimes
\Omega^\bullet M$; namely, if we consider maps
$$f: G^k \to S{\mathfrak g}^* \otimes
\Omega^\bullet M$$ with image homogeneous elements of degree $l$,
we will denote $deg_1(f)=k$ and $deg_2(f) = l$. Then we can define
the filtration
$$F^p:= \{ f \in C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M | deg_2(f)\geq p \}$$ where  $F^{p+1} \subset
F^p$. We have that the differentials have the following degrees:
\begin{align*}
deg_1(d)& =0 &  deg_2(d)& =1\\
deg_1(\iota)&=0  & deg_2(\iota)& =1\\
deg_1(\overline{d})&=1 & deg_2(\overline{d})&=0\\
deg_1(\overline{\iota})&=-1& deg_2(\overline{\iota})&=2\end{align*}
and therefore the filtration is compatible with the differentials.

The spectral sequence associated to the filtration $F^*$ has for
page 0:
$$E_0 = \bigoplus_p F^p/F^{p+1} \cong C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M )$$ and the 0-th differential is $d_0 =
\overline{d}$ because the other three differentials raise $deg_2$.
Therefore the page 1 is:
$$E_1 = H^*(C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M ), \overline{d})$$ the differentiable cohomology of
$G$ with coefficients in the representation $S{\mathfrak g}^* \otimes
\Omega^\bullet M$. The 0-th row of the first page is precisely the Cartan complex
$$E_1^{*,0} = H^0(C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M ), \overline{d}) = (S{\mathfrak g}^* \otimes
\Omega^\bullet M)^G =C_G(M)$$
and the first differential on this row  $d_1 : E_1^{*,0} \to E_1^{*+1,0}$ becomes precisely the Cartan differential $d + \iota$. Therefore we see that on the second page we get that
$$E_2^{*,0} \cong H^*(C_G(M), d+ \iota),$$
namely that the 0-th row of the second page is isomorphic to the cohomology of the Cartan complex.

Therefore we conclude that the composition
$$E_2^{*,0} \to H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$$
of the surjective homomorphism $E_2^{*,0}  \to E_\infty^{*,0} $ with the inclusion
$$E_\infty^{*,0} \subset H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$$
is equivalent to the induced map on cohomologies
$$i : H^*(C^*_G(M), d+\iota) \to H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M
)$$
defined previously.

\subsection{Differentiable cohomology of Lie groups}

Notice that in De Rham model for equivariant cohomology the operator $\overline{d}$ is defined in a similar fashion
as the differential for group cohomology in the case of discrete groups. The cohomology groups defined by the differential $\overline{d}$
are called the {\it differentiable cohomology groups} and are defined for any $G$-module $V$; i.e. if $G$ is a Lie group and $V$ is a $G$-module then the {\it differentiable cohomology of $G$ with values
in $V$} is the cohomology of the complex $C_d^*(G,V)$ where $C_d^k(G,V)$ consists of differentiable maps $f: G^k \to V$ such
that $f$  vanishes if any of the arguments $g_i$ equals the identity of
$G$, and the differential is $\overline{d}$ is defined by
\begin{eqnarray*}
(\bar{d}f)(g_0, \dots , g_k) & = & f( g_1, \dots , g_k ) +
 \sum_{i=1}^k (-1)^i f(g_0, \dots, g_{i-1}g_i, \dots  , g_k )\\
 & & +(-1)^{k+1} g_k f(g_0, \dots , g_{k-1} ).
\end{eqnarray*}
We denote this cohomology by
$$H_d^*(G,V) := H^*( C^*(G,V), \overline{d}).$$

For $V$ in the category of topological $G$-modules, the cohomology groups $H_d^*(G,V)$ can be seen as the relative derived functor associated
to the $G$ invariant submodule $V^G$. In particular we have that
$$H_d^0(G,V)=V^G,$$
and whenever $G$ is compact the functor of taking the $G$-invariant submodule is exact and therefore
in that case $H_d^{*>0}(G,V)=0$.

\begin{rem}
The first page of the spectral sequence that was defined in section \S \ref{subsubsection spectral sequence}
$$E_1 = H^*(C^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M ), \overline{d})$$
is precisely the differential cohomology defined above
$$E_1= H_d^*(G, S{\mathfrak g}^* \otimes
\Omega^\bullet M )$$
for the $G$-module $S{\mathfrak g}^* \otimes
\Omega^\bullet M$.
\end{rem}

For $V$ a vector space over $\ensuremath{{\mathbb R}}$, by differentiating the functions from $G^k$ to $V$, Van Est \cite{VanEst1} proved that for $G$ connected
$$H_d^*(G;V) \cong H^*(\mathfrak{g},\mathfrak{k};V)$$
whenever $K$ is the maximal compact subgroup of $G$, $\mathfrak{k}$ and $\mathfrak{g}$ are their corresponding Lie algebras, and
$H^*(\mathfrak{g},\mathfrak{k};V)$ denotes the Lie algebra cohomology defined by Chevalley and Eilenberg in \cite{ChevalleyEilenberg}.

Whenever the group $G$ has a compact form $G_u$, i.e. a compact Lie group whose complexification is isomorphic to the complexification
of $G$
$$G_\ensuremath{{\mathbb C}} \cong (G_u)_\ensuremath{{\mathbb C}},$$
then
we have that
$$H^*(\mathfrak{g},\mathfrak{k}; V) \cong H^*(\mathfrak{g}_u,\mathfrak{k}; V) \cong H^*(G_u/K;V),$$
where the second isomorphism was proved by Chevalley and Eilenberg in \cite{ChevalleyEilenberg} for compact Lie groups,
and the first isomorphism follows from the isomorphisms
$$H^*(\mathfrak{g},\mathfrak{k}; V) \otimes_\ensuremath{{\mathbb R}} \ensuremath{{\mathbb C}} \cong H^*(\mathfrak{g}_\ensuremath{{\mathbb C}},\mathfrak{k}_\ensuremath{{\mathbb C}}; V_\ensuremath{{\mathbb C}}) \cong
H^*(\mathfrak{g}_u,\mathfrak{k}; V) \otimes_\ensuremath{{\mathbb R}} \ensuremath{{\mathbb C}} ,$$ which follow from
the isomorphism of complex Lie algebras $\mathfrak{g}_\ensuremath{{\mathbb C}} \cong
(\mathfrak{g}_u)_\ensuremath{{\mathbb C}}$. In this case we have that the differentiable
cohomology of $G$ can be calculated by topological methods, i.e.
$$H_d^*(G;V) \cong H^*(G_u/K; V).$$

\subsubsection{Example $G=SL(n,\ensuremath{{\mathbb R}})$} \label{section differentiable cohomology of SL(n,R)} Let us
consider the non-compact group $G=SL(n,\ensuremath{{\mathbb R}})$.
In this case we have \begin{align*}
\mathfrak{g}&=\mathfrak{sl}(n,\ensuremath{{\mathbb R}}) &
\mathfrak{g}_\ensuremath{{\mathbb C}}&= \mathfrak{sl}(n,\ensuremath{{\mathbb C}})\\
\mathfrak{g}_u&= \mathfrak{su}(n) &
\mathfrak{k}&= \mathfrak{so}(n)\\
\mathfrak{k}_\ensuremath{{\mathbb C}}&=\mathfrak{so}(n,\ensuremath{{\mathbb C}}) &
K&=SO(n)\\
G_u&=SU(n). & &\end{align*}
Therefore for $n>2$ we have that
$$H_d^*(SL(n,\ensuremath{{\mathbb R}}),\ensuremath{{\mathbb R}}) = H^*(SU(n)/SO(n); \ensuremath{{\mathbb R}}) =\Lambda[h_3,h_5,...,h_{\langle n \rangle}],$$
where the degree of $h_i$ is $2i-1$ and $\langle n \rangle$ is the largest odd integer which is less or equal than $n$.

The equivariant De Rham cohomology of this group is $H^*(G,S\mathfrak{g}^*)$. The spectral sequence defined in section
\S \ref{subsubsection spectral sequence} has  for first page
$$E_1 = H_d^*(G, S\mathfrak{g}^*) \cong H^*(\mathfrak{g},\mathfrak{k}; S\mathfrak{g}^*)$$
and since the algebra $\mathfrak{g}=\mathfrak{sl}(n,\ensuremath{{\mathbb R}})$ is reductive and $S\mathfrak{g}^*$ is a finite dimensional semi-simple $\mathfrak{g}$-module in each degree \cite{Berger-et-al}, then we have that
$$H^*(\mathfrak{g},\mathfrak{k}; S\mathfrak{g}^*) \cong H^*(\mathfrak{g},\mathfrak{k}; \ensuremath{{\mathbb R}}) \otimes (S\mathfrak{g}^*)^{G}$$
and therefore
$$E_1 \cong H_d(G;\ensuremath{{\mathbb R}}) \otimes (S\mathfrak{g}^*)^G.$$

The ideal of $G$-invariant polynomials  is known to be
$$(S\mathfrak{g}^*)^G \cong \ensuremath{{\mathbb R}}[c_2,c_3,...,c_n],$$
where the degree of $c_i$ is $2i$, and so we get that the first page of the spectral sequence
converging to $H^*(G,S \mathfrak{g}^*)$ is
$$E_1 \cong \Lambda[h_3,h_5,...,h_{\langle n \rangle}] \otimes \ensuremath{{\mathbb R}}[c_2,c_3,...,c_n].$$

Since we know that  for $n>2$
$$H^*(G,S\mathfrak{g}^*)= H^*(BG;\ensuremath{{\mathbb R}}) = H^*(BSO(n);\ensuremath{{\mathbb R}}) \cong \ensuremath{{\mathbb R}}[c_2,c_4,...c_{2[n/2]}] $$
is the free algebra on the Pontrjagin classes, we obtain that the
$(2i)$-th differential of the spectral sequence maps the class $h_i$
to the class $c_i$ $$h_i \stackrel{d_{2i}}{\mapsto} c_i.$$

In particular we obtain that
\begin{equation} \label{fourth cohomology of BSL(n,R)}
\ensuremath{{\mathbb R}}=H^4(G,S\mathfrak{g}^*) \cong E_\infty^{4,0} \cong E_1^{4,0} = (S^2 \mathfrak{g}^*)^G,\end{equation}
and therefore we see that the fourth cohomology group of $BG$ is generated by
the $G$-invariant quadratic forms in  $(S^2 \mathfrak{g}^*)^G$.

\section{Equivariant extension of differential forms} \label{section extension differential form}

In many instances in geometry, the action of a compact Lie group on a manifold being of a certain kind is equivalent to the existence of an equivariant lift
 of a specific invariant closed differential form
on the Cartan model. Some examples of this phenomenon are the following:
\begin{itemize}
\item The action of $G$ on a symplectic manifold $(M,\omega)$ being Hamiltonian is equivalent to the existence of
a closed equivariant lift $\tilde{\omega}= \omega + \mu_a\Omega^a \in C_G^2(M)$ of the symplectic form; in this case the maps $\mu_a: M \to \ensuremath{{\mathbb R}}$
may be assembled into a map $\mu: \mathfrak{g} \to C^\infty M$, $\mu(a):=\mu_a$, which becomes the moment map (see \cite{AtiyahBott}).
\item The action of $G$ on an exact Courant algebroid $(TM \oplus T^*M, [,]_H)$, with Courant-Dorfman bracket twisted by the closed three form $H$, is Hamiltonian provided there exists a closed equivariant lift $\tilde{H}= \omega + \xi_a\Omega^a \in C_G^3(M)$ (see \cite{Bursztyn, CaviedesHuUribe, Hu2, Uribe-dg}).
\item Let $\Gamma$ be a connected, simple, simply connected and compact matrix group, i.e.  $\Gamma \subset GL(N,\ensuremath{{\mathbb R}})$, and denote by $\Gamma_L \times \Gamma_R$ the product
of two copies of $\Gamma$ acting on $\Gamma$ on the left by the action
\begin{align*}
(\Gamma_L \times \Gamma_R) \times \Gamma & \to \Gamma \\
((g,h),k) & \mapsto gkh^{-1}.
\end{align*}
A subgroup $G \subset \Gamma_L \times \Gamma_R$ is called an {\it
anomaly-free subgroup} if there the WZW action could be gauged with
respect to the group action given by $G$. In \cite{Witten} Witten
showed that the anomaly-free subgroups are precisely the subgroups
$G$ on which the WZW term
$$\omega = \frac{1}{12 \pi} {\rm{Tr}}(g^{-1} dg)^3$$
could be lifted to a closed equivariant 3-form in the Cartan complex $\tilde{\omega}= \omega - \lambda_a \Omega^a \in C^3_G(\Gamma)$.
(We will elaborate this construction in the next chapter).
\end{itemize}

The previous examples are not exhaustive, but they give the idea of
the general principle. In these cases we have a $G$-invariant closed
form $\omega$ and we need to find a closed equivariant lift
$\tilde{\omega}$. Note that since the group $G$ is compact, the
existence of a closed equivariant lift in the Cartan model is
equivalent to the existence of a lift of the cohomology class
$[\omega] \in H^*(M; \ensuremath{{\mathbb R}})$ on the equivariant cohomology $H^*(EG
\times_G M; \ensuremath{{\mathbb R}})$. Therefore the obstructions of the existence of
the equivariant lift could be studied via several methods, for
instance, with the use of the Serre spectral sequence associated to
the fibration $M \to EG \times_G M \to BG$, or with the use of the
spectral sequence associated to a filtration of the Cartan complex
$C_G(M)$ given by the degree of $S \mathfrak{g}^*$ as it is done in the papers
\cite{Figueroa, Wu}.

But what happens in the case that the Lie group $G$ is not compact?
We speculate that the situation should be similar, in the sense that
the action being of certain kind is equivalent to the existence of a
lift on the equivariant De Rham complex defined in section
\S\ref{equivariant De Rham complex}. This situation has not been
explored so far but we believe that the equivariant De Rham complex
provides a framework in which actions on non-compact Lie groups
could be better understood.

Since in many instances the geometric information is captured by
closed forms in the Cartan complex, we would like to study the
relation between the cohomology of the Cartan complex and the
equivariant De Rham cohomology whenever the group is not compact.

\subsection{Cartan complex vs. equivariant De Rham complex}

Let us consider the diagram of complexes
$$\xymatrix{
C^*_G(M) \ar[d] \ar[r]^(.35){i} & C^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M
) \ar[ld] \\
\Omega^\bullet M & }$$
where the horizontal  map is the injective map of complexes defined in \eqref{homomorphism Cartan to De Rham}
and the vertical maps are the natural forgetful maps. Let us take a closed $G$-invariant form $H$ on $M$ and let us suppose that
we can lift this closed form to a closed form in the Cartan model $\overline{H} \in C_G(M)$, then equivariant form $i\overline{H}$
becomes a closed lift for $H$ in the complex of equivariant De Rham forms, i.e. $i\overline{H} \in
 C^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$. We have then

 \begin{lemma} \label{extension in Cartan implies equivariant extension}
 Take a closed $G$-invariant form $H \in (\Omega^\bullet M)^G$. If $H$ can be lifted to a closed form $\overline{H}$ in the Cartan model of the equivariant
 cohomology, then the form $i(\overline{H})$ is a closed lift for $H$ in $ C^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$, the closed forms of the
 equivariant De Rahm complex.
 \end{lemma}

Note in particular that Lemma \ref{extension in Cartan implies equivariant extension} implies that if one can extend an invariant closed form
to a closed form in the Cartan model for equivariant cohomology, then the cohomology class $[H]$ lies in the image of the canonical
forgetful homomorphism
$$H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M) \to H^*(M),$$
and therefore the cohomology class $[H]$ could be extended to an equivariant cohomology class in any model for the equivariant cohomology of $M$.

The converse of Lemma  \ref{extension in Cartan implies equivariant extension}  would say
that if one knows that an invariant differentiable form $H$ could be lifted to a closed form in $ C^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$
then a lift could be written as a closed element in the Cartan model. The converse of Lemma  \ref{extension in Cartan implies equivariant extension}  is indeed true whenever the Lie group $G$ is compact, but
for general group actions it does not hold.

From the spectral sequence defined in section \S \ref{subsubsection spectral sequence} we have seen that in the first page we have
$$E_1^{*,0} \cong (S \mathfrak{g}^* \otimes \Omega ^\bullet M)^G = C_G(M)$$
with differential $d_1 : E_1^{*,0} \to E_1^{*+1,0}$ equivalent to $d + \iota$. Therefore on the second page we get that
\[E_2^{*,0} \cong H^*(C_G(M), d+ \iota),\]
namely that the 0-th row of the second page is isomorphic to the cohomology of the Cartan complex.

 Since we have the surjective homomorphism $E_2^{*,0}  \to E_\infty^{*,0} $ we can conclude that

 \begin{proposition} \label{conditions extension in Cartan}
 For a closed $G$-invariant form $H \in (\Omega^\bullet M)^G$, it can be lifted to a closed form in the Cartan complex if,
 firstly the cohomology class $[H]$ could be lifted to an equivariant cohomology class in $H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$, and secondly, if the lift lies on the subgroup $E_\infty^{*,0} \subset H^*(G, S{\mathfrak g}^* \otimes \Omega^\bullet M)$.
 \end{proposition}

 The second condition of Proposition \ref{conditions extension in Cartan} is more difficult to check than the first one, since it depends
 explicitly on the equivariant De Rham model for equivariant cohomology; for the first condition any model for the equivariant cohomology
 works.
 In certain specific situations, extensions in the Cartan model of closed forms may be obtained, and this is the subject of the next and final chapter.

 We note here that a sequence of obstructions for lifting a $G$-invariant form to a closed form in the Cartan complex can
 be determined, when studying the spectral sequence associated to appropriate filtrations of the Cartan complex, as it is carried out in
 \cite{Figueroa, Wu}. Our approach is different since we are interested in using the fact that an extension in the Cartan model can only
 exist if there is an equivariant extension, i.e. an extension in the homotopy quotient.

 \section{Equivariant extensions of the WZW term for $SL(n,\ensuremath{{\mathbb R}})$ actions}

 In the physics literature (see \cite{Witten} and the references therein) it has been argued that the
 condition of anomaly cancelation for the gauged WZW action is given by the equation
 \begin{align*}
 {\rm{Tr}}(T_{a,L} T_{b,L}- T_{a,R} T_{b,R})=0
 \end{align*}
and this equation is moreover equivalent to the existence of an equivariant extension of the WZW term
 $$\omega = \frac{1}{12 \pi} {\rm{Tr}}(g^{-1} dg)^3$$
 on the Cartan model for equivariant cohomology. In the case that the group that we are gauging is compact, the anomaly cancellation is  equivalent to the existence of an equivariant lift of the cohomology class $[\omega]$, and therefore the anomaly cancellation becomes topological and could be checked with topological methods.

 In this section we study in detail the case on which the gauge group is $G=SL(n,\ensuremath{{\mathbb R}})$ (or any subgroup of it)
 and we show that the anomaly cancelation condition for the WZW action is also topological and only depends on the existence
 of an equivariant lift of the cohomology class $[\omega]$; in this way we find a large family of $SL(n,\ensuremath{{\mathbb R}})$ actions with anomaly cancellation.

 Let us start by recalling the explanation given by Witten \cite{Witten} that asserts that the
 condition for anomaly cancelation is equivalent to the existence of a lift in the Cartan complex of the form
$\omega$.

 \subsection{Gauged WZW actions} \label{section gauged WZW actions}

  Let $\Gamma$ be a connected and simple matrix group, i.e.  $\Gamma \subset GL(N,\ensuremath{{\mathbb R}})$, such that its fundamental group
  is finite.
  Denote by $\Gamma_L \times \Gamma_R$ the product
of two copies of $\Gamma$ acting on $\Gamma$ on the left by the action
\begin{align*}
(\Gamma_L \times \Gamma_R) \times \Gamma & \to \Gamma \\
((g,h),k) & \mapsto gkh^{-1}
\end{align*}
and consider a subgroup $G \subset \Gamma_L \times \Gamma_R$  acting on the left on $\Gamma$ by the induced action of
$\Gamma_L \times \Gamma_R$.

The embedding $G \subset \Gamma_L \times \Gamma_R$ determines a map at the level of Lie algebras which can be written
as $$a \mapsto (T_{a,L}, T_{a,R}), \ \ a \in \mathfrak{g}$$
and the canonical vector fields $X_a$ on $\Gamma$ generated by the $G$ action could be written as
$$(X_a)_g = T_{a,L} g-g T_{a,R}$$
  for all $g \in \Gamma$.

  The matrix 1-forms $g^{-1}dg$ and $dg  g^{-1}$ satisfy the equations
  \begin{align*}
  \iota_{X_a} (g^{-1}dg) & = g^{-1} T_{a,L} g - T_{a,R},\\
   \iota_{X_a}(dg g^{-1}) & =   T_{a,L} - g^{-1} T_{a,R} g,\\
  d(dg g^{-1})^{2p+1} &=  -(dg g^{-1})^{2p+2},\\
  d(g^{-1}dg)^{2p+1} &=  ( g^{-1}dg)^{2p+2}
  \end{align*}
and we can take the differential form
$$\omega = \frac{1}{12 \pi} {\rm{Tr}}(g^{-1} dg)^3,$$
which defines the WZW action. The form $\omega \in \Omega^3 \Gamma$ is $\Gamma_L \times \Gamma_R$ invariant, therefore
$\omega \in (\Omega^3 \Gamma)^G$,  it is closed and is a generator of the cohomology group $H^3(\Gamma)=\ensuremath{{\mathbb R}}$.

To find a closed extension of $\omega$ in the Cartan complex we need
to find 1-forms $\lambda_a$ such that the following equations are
satisfied:
\begin{align*}
\iota_{X_a} \omega - d \lambda_a=0, \\
\iota_{X_a} \lambda_b + \iota_{X_b} \lambda_a=0,\\
\mathcal{L}_{X_b} \lambda_a = \lambda_{[b,a]},
\end{align*}
where the first two imply that the form $\tilde{\omega}=\omega - \lambda_a \Omega^a$ is $(d+\iota)$-closed, and the third one
implies that $\tilde{\omega}$ is $G$-invariant. Calculating $\iota_{X_a} \omega$ we obtain
\begin{align*}
\iota_{X_a} {\rm{Tr}}(g^{-1} dg)^3 &= 3 {\rm{Tr}}\left((g^{-1} T_{a,L} g - T_{a,R}) (g^{-1}dg)^2 \right)\\
& = 3{\rm{Tr}}\left( T_{a,L} (dg g^{-1})^2 - T_{a,R} (g^{-1}dg)^2 \right)\\
&= d \left[3{\rm{Tr}}\left( T_{a,L} (dg g^{-1}) +T_{a,R} (g^{-1}dg) \right) \right]
\end{align*}
 and therefore we see that we can define
 $$\lambda_a= \frac{1}{4 \pi} {\rm{Tr}}\left( T_{a,L} (dg g^{-1}) +T_{a,R} (g^{-1}dg) \right)$$
 satisfying the equation $\iota_{X_a} \omega - d \lambda_a=0$.
  The fact that $\mathcal{L}_{X_b} \lambda_a = \lambda_{[b,a]}$ is satisfied, is a tedious but straightforward computation.

 Now we compute
 \begin{align*}
\iota_{X_a} \lambda_b + \iota_{X_b} \lambda_a &= \frac{1}{2 \pi} {\rm{Tr}} \left( T_{a,L}T_{b,L} - T_{a,R}T_{b,R} \right)
 \end{align*}
noting that for $a,b \in \mathfrak{g}$ the function thus defined become constant, and therefore we have that
 $$\tilde{\omega}=\omega - \lambda_a \Omega^a \in (S \mathfrak{g}^* \otimes \Omega^\bullet \Gamma)^G$$
 is a 3-form in the Cartan complex $C_G(\Gamma)$ and its failure to be closed is the quadratic form
 \begin{align*}
 (d + \iota) \tilde{\omega} \in (S^2 \mathfrak{g}^*)^G,
 \end{align*}
 where the coefficient of $\Omega^a \Omega^b$ of this quadratic form is precisely
 $$-\frac{1}{2 \pi} {\rm{Tr}} \left( T_{a,L}T_{b,L} - T_{a,R}T_{b,R} \right).$$

Since it was known in the literature that the condition for the absence of anomalies was
$${\rm{Tr}} \left( T_{a,L}T_{b,L} - T_{a,R}T_{b,R} \right)=0,$$
Witten concluded that the absence of anomalies was equivalent to the existence of a closed extension of $\omega$ in the Cartan complex.

Whenever the Lie group $G$ is compact, the existence of such an extension is equivalent to the existence of an equivariant
extension of the cohomology class $[\omega]$, and this can be checked with the use of the Serre spectral sequence associated
to the fibration
$$\Gamma \to EG \times_G \Gamma \to BG.$$
The second page of the spectral sequence becomes
$$E_2^{p,q}= H^p(BG; H^q(\Gamma ;\ensuremath{{\mathbb R}})) \cong H^p(BG;\ensuremath{{\mathbb R}}) \otimes H^q(\Gamma ;\ensuremath{{\mathbb R}}),$$
and since $H^1(\Gamma ;\ensuremath{{\mathbb R}})=H^2(\Gamma ;\ensuremath{{\mathbb R}})=0$, the only
non-trivial differential that affects $[\omega] \in E_2^{0,3}$ is
$d_4$ thus defining an element
$$d_4([\omega]) \in E_2^{4,0} \cong H^4(BG;\ensuremath{{\mathbb R}});$$
this implies that the only obstruction to lift $[\omega]$ to an equivariant class is precisely $d_4([\omega])$.
Since we assumed that $G$ is compact, we know that
$$H^4(BG;\ensuremath{{\mathbb R}}) \cong (S^2 \mathfrak{g}^*)^G$$
and therefore we must have that
$$d_4([\omega]) = (d+\iota)\tilde{\omega},$$
namely that the two obstructions are the same.

The previous argument permits to find several cases on which there is anomaly cancellation. The simplest of all is the adjoint
action of $G$ on itself $\Gamma=G$ since in this case the spectral sequence associated to the fibration $EG \times_G G_{ad} \to BG$
always collapses at the second page, and therefore $d_4=0$.

 \subsection{WZW actions with gauge group $G=SL(n,\ensuremath{{\mathbb R}})$}
 In this section we will argue that if the gauge group is $G=SL(n,\ensuremath{{\mathbb R}})$ then the cancellation of anomalies is topological, and therefore
 it is equivalent to  the existence of an equivariant lift of the cohomology class $[\omega]$.

 \begin{theorem} \label{equivalence of extension for SL(n,R)}
 Let $\Gamma$ be a connected, simple matrix group with finite fundamental group.
 Let $G=SL(n,\ensuremath{{\mathbb R}})$ and consider an action on $\Gamma$ defined by an injection $G \subset \Gamma_L \times \Gamma_R$.
 The existence of an equivariant extension on $H^3(EG \times_G \Gamma; \ensuremath{{\mathbb R}})$ of the cohomology class $[\omega]$ is equivalent
 to the existence of a closed lift to the Cartan model of $\omega$.
  \end{theorem}

\begin{proof}
We know that if there is an extension $\tilde{\omega}$ in the Cartan model, then the cohomology class $[\tilde{\omega}]$ represents
the lift in the cohomology group $H^3(G, S \mathfrak{g}^* \otimes \Omega^\bullet \Gamma)$. To prove the converse we will make use of the constructions and results of sections  \S \ref{section differentiable cohomology of SL(n,R)}, \S \ref{section extension differential form} and \S\ref{section gauged WZW actions}.

From the equivariant De Rham theorem we know that $H^*(BG;\ensuremath{{\mathbb R}}) \cong H(G, S \mathfrak{g}^*)$ and therefore we could
take the class $d_4([\omega])$, which is the obstruction of extending $[\omega]$ to an equivariant class, to be an element in
$$d_4([\omega]) \in H^4(G, S \mathfrak{g}^*).$$

We already know that $(d+ \iota) \tilde{\omega} \in (S^2 \mathfrak{g}^*)^G$ and its cohomology class in $H^4(G, S \mathfrak{g}^*)$ represents
the same obstruction for an equivariant lift, i.e.
$$d_4([\omega]) = [(d+ \iota) \tilde{\omega}].$$
In general it may happen that the cohomology class $ [(d+ \iota) \tilde{\omega}]$ is zero even though the form $(d+ \iota) \tilde{\omega}$
is different from zero. But in the particular case of $G=SL(n, \ensuremath{{\mathbb R}})$ we have already seen in \eqref{fourth cohomology of BSL(n,R)}
that the inclusion map of the Cartan complex into the equivariant De Rham complex
$$(S\mathfrak{g}^*)^G \to C^*(G, S\mathfrak{g}^*)$$
induces an isomorphism in degree 4
$$(S^2\mathfrak{g}^*)^G \stackrel{\cong}{\to} H^4(G,S\mathfrak{g}^*), \ \ (d+\iota)\tilde{\omega} \mapsto [(d+\iota)\tilde{\omega}]$$
and therefore we have that the vanishing of the class $d_4([\omega])$ is equivalent to the vanishing of the quadratic form $(d+\iota)\tilde{\omega}$, i.e.
$$d_4([\omega])=0 \ \ \mbox{if and only if} \ \ (d+\iota)\tilde{\omega}=0.$$
\end{proof}

We see that for the case on which the gauge group is $SL(n,\ensuremath{{\mathbb R}})$, the equations of cancellation of anomalies, namely that
for all $a,b \in \mathfrak{sl}(n,\ensuremath{{\mathbb R}})$
$${\rm{Tr}} \left( T_{a,L}T_{b,L} - T_{a,R}T_{b,R} \right)=0,$$
are equivalent to the existence of an equivariant extension on $H^3(EG \times_G \Gamma; \ensuremath{{\mathbb R}})$ of the cohomology class
$[\omega]$. Now we are ready to give examples on both the existence and the non existence of equivariant extensions of $\omega$.

\subsection{Examples}
\subsubsection{Adjoint action}

Let $G=\Gamma$ and consider the diagonal injection
$$G  \subset \Gamma_L \times \Gamma_R, \ \ g \mapsto (g,g)$$
which induces the adjoint action of $G$ on $\Gamma=G^{\rm{ad}}$. In this case the cohomology of the homotopy quotient  $EG \times_G G^{\rm{ad}}$ is isomorphic to the cohomology of $G$ tensor the cohomology of $BG$:
$$H^*(EG \times_G G^{ad}; \ensuremath{{\mathbb R}})\cong   H^*(BG;\ensuremath{{\mathbb R}}) \otimes H^*(G;\ensuremath{{\mathbb R}}).$$
This isomorphism can be obtained from the Serre spectral sequence associated to the fibration
$$G \to EG \times_G G^{\rm{ad}} \to BG,$$
which collapses at level 2 because the classes in $H^*(G;\ensuremath{{\mathbb R}})$
can be lifted to classes to $H^*(EG \times_G G^{ad}; \ensuremath{{\mathbb R}})$: take
a primitive class in $H^*(BG;\ensuremath{{\mathbb R}})$ (namely a class in
$H^*(BG;\ensuremath{{\mathbb R}})$ which is in the image of a primitive element in
$H^*(G)$ of one of the differentials of the Serre Spectral
Sequence associated to the fibration $G \to EG \to BG$) pull it
back to $S^1 \times \mathcal{L} BG$ via the evaluation map where $\mathcal{L} BG$
is the space of free loops of $BG$, then integrate over $S^1$ and
get a class in $H^*(\mathcal{L} BG;\ensuremath{{\mathbb R}})$ of degree 1 less; this class in
$H^*(\mathcal{L} BG;\ensuremath{{\mathbb R}})$ once restricted to the based loops $\Omega BG
\simeq G$ is precisely the class in $H^*(G)$ that defined the
primitive class in $H^*(BG)$ that we started with; finally recall
that $\mathcal{L} BG \simeq EG \times_G G^{ad}$.

If we take $G=\Gamma=SL(n,\ensuremath{{\mathbb R}})$, we know by Theorem
\ref{equivalence of extension for SL(n,R)} that the existence of an
equivariant extension of $[\omega]$ is equivalent to the cancelation
of anomalies for this gauged action, and since the class $[\omega]$
can be extended to an equivariant one, we conclude that in this case
there is an anomaly cancelation.

In particular, for any subgroup $F \subset SL(n,\ensuremath{{\mathbb R}})$ acting by
the adjoint action on $SL(n,\ensuremath{{\mathbb R}})$ there is also cancelation of
anomalies.

\subsubsection{$G=SL(n,\ensuremath{{\mathbb R}}) \subset \Gamma_L$ for $n>2$ }

Whenever the action of $G=SL(n,\ensuremath{{\mathbb R}})$ on $\Gamma$ is obtained by a left action induced by an inclusion $G=SL(n,\ensuremath{{\mathbb R}}) \subset \Gamma_L$, we have that the $G$ action on $\Gamma$ is free and therefore the homotopy quotient $EG \times_G \Gamma$ and the quotient $G \backslash \Gamma$ are homotopy equivalent. For $n>2$ the Serre spectral sequence tells us that $$d_4([\omega])=c_2 \in H^4(BSL(n,\ensuremath{{\mathbb R}});\ensuremath{{\mathbb R}})$$
and therefore the class $[\omega]$ does not extend to an equivariant one. By Theorem \ref{equivalence of extension for SL(n,R)} we know that this implies that there is no cancellation of anomalies. We conclude that for left free actions of the group $SL(n,\ensuremath{{\mathbb R}})$ for $n>2$  there must exist $a,b \in \mathfrak{sl}(n,\ensuremath{{\mathbb R}})$ such that
$${\rm{Tr}} \left( T_{a,L}T_{b,L} - T_{a,R}T_{b,R} \right)\neq 0.$$

\subsubsection{$G=SL(2,\ensuremath{{\mathbb R}}) \subset \Gamma_L$ }

Following the same argument as before, we have that
$d_4([\omega])=0$ since $H^4(BSL(2,\ensuremath{{\mathbb R}}),\ensuremath{{\mathbb R}})=0$. Therefore for
left free actions of the group $SL(2,\ensuremath{{\mathbb R}})$ there is anomaly
cancelation.

\def$'${$'$}

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RKSV De Tukkers is een omnisportvereniging uit Albergen, gemeente Tubbergen, Overijssel, Nederland.

Algemeen
De vereniging werd op 14 september 1932 opgericht en telt afdelingen voor voetbal en handbal. De thuiswedstrijden worden op het "Sportpark 'T Loarkamp" en in de sportzaal "De Danne" gespeeld.

Geschiedenis
1932 - Het zijn de pioniersjaren voor De Tukkers. Vijf jongens die het voortouw nemen en TVC'28 de rug toedraaien. Ze zijn het beu nog langer naar Tubbergen te fietsen om een potje te voetballen. Ze willen een eigen vereniging. De bij TVC spelende Albergenaren gaan voor advies naar enkele ouderen. Er moet namelijk een bestuur komen en dat heeft nogal wat voeten in aarde.

Nadat er de goedkeuring is ontvangen van het bisdom Utrecht, komt men begin september 1932 een paar avonden bij elkaar om het een en ander te bespreken. De uiteindelijke oprichtingsvergadering wordt belegd op 14 september 1932 in Café Morshuis.

De wei, bestaande uit lage, kleiachtige grond wordt door Morshuis beschikbaar gesteld. De ligging is de plaats waar op dit moment ons B-veld gesitueerd is. De ingang van het veld is aan de achterzijde en bereikbaar via een pad vanuit Café Morhuis. Morshuis zelf verzorgt de weide en stelt de goalpalen beschikbaar, terwijl de leden zorgen voor vier zwart-witte hoekvlaggen en zand om het speelveld af te bakenen.

Op het speelveld blijft echter wel het vee grazen, zodat voor elke wedstrijd eerst de kötteltöp gemaaid moeten worden en de koeienplakken met de bats van het veld worden gehaald. De schuur van Morshuis wordt gebruikt als kleedruimte.

De Tukkers wordt onder de naam R.K.V. De Tukkers ingeschreven bij de Roomsch Katholieke Utrechtsche Voetbalbond en ingedeeld in de 3e klasse E. Later zal de naam veranderen in R.K.S.V. De Tukkers. Op 19 september zegent de pastoor het veld, waarna voetbalminnend Albergen met spanning afwacht op de eerste verrichtingen van hun nieuwbakken voetbalteam.

Voetbal

Standaardelftal
Het standaardelftal speelt in het seizoen 2022/23 in de Tweede klasse zondag van het KNVB-district Oost.

Competitieresultaten 1957–2023

 2004: de beslissingswedstrijd op 6 mei bij Quick '20 om het klassekampioenschap in zondag 4B werd met 1-1 gelijkgespeeld tegen DSVD, De Tukkers kampioen op basis beter doelsaldo in competitie.

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Erelijst
kampioen Derde klasse: 2022

kampioen Vierde klasse: 2015

kampioen Vierde klasse: 2004

kampioen Derde klasse: 1998

kampioen Vierde klasse: 1996

kampioen Derde klasse: 1993

kampioen Vierde klasse: 1991

kampioen Vierde klasse: 1972

kampioen Vierde klasse: 1965

kampioen Vierde klasse: 1957 (geen promotie)

Vrouwen
Het eerste vrouwenvoetbalelftal speelt in het seizoen 2022/23 in de landelijke Hoofdklasse zondag. In het seizoen 2018/19 werden ze kampioen in de Tweede klasse, in 2017/18 in de Derde klasse.

Erelijst

kampioen Tweede klasse: 2019
kampioen Derde klasse: 2018

Handbal
Het eerste vrouwenteam werd in 2019 kampioen en spelen sinds het seizoen 2019/20 in de landelijke 1e divisie.

De topscorer aller tijden van het Nederlands handbalteam, Olga Assink, begon haar carrière bij De Tukkers. Van 2013 tot 2020 is ze trainster geweest van het eerste vrouwenteam.

Erelijst
kampioen Tweede divisie: 2019

Amateurvoetbalclub in Overijssel
RKSV
Nederlandse omnisportvereniging
Nederlandse handbalclub

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