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After a last-gasp penalty against Leicester City helped Liverpool make it eight wins from eight games on Saturday, City were expected to respond but instead went down 2-0 with Adama Traore scoring a late double at a stunned Etihad Stadium.
Pierre-Emerick Aubameyang's 58th minute goal for the visitors, given after a VAR review, cancelled out Scott McTominay's well-struck opener on the stroke of halftime but it was a game which highlighted the weaknesses of the teams.
West Ham have 11 points after six games, the same as third-placed Leicester City, while Manchester United are eighth on eight points.
Manchester United earned a first victory since the opening weekend by beating Leicester City 3-1, while Chelsea and Tottenham Hotspur registered emphatic wins.
United, who were without injured French internationals Paul Pogba and Anthony Martial, were far from convincing but manager Ole Gunnar Solkskjaer was delighted they secured their second win of the campaign to reach eight points from five games.
Leicester moved up to third as the Foxes unbeaten start to the season continued in a 3-1 win over Bournemouth.
The result left both sides on four points from three games.
After a 4-0 home win over Chelsea on the opening day, Ole Gunnar Solskjaer's team dominated the early exchanges and led after 27 minutes when Marcus Rashford slipped the ball in for Martial to score his 50th goal for the club in all competitions.
The win was United's biggest over Chelsea in a top-flight match since the same scoreline in March 1965 under Matt Busby.
Tottenham's north London rivals Arsenal were also active, spending a reported 25 million pounds on Celtic's highly-rated Scotland left-back Kieran Tierney.