Manchester City lost their cool in a damaging defeat against title rivals Chelsea, Liverpool surrendered ground on the leaders after an incredible collapse at Bournemouth, while Alexis Sanchez kept Arsenal firmly in the hunt.

Here are five things we learned in the Premier League this weekend:

Man City need to regain composure

It is saying something when the calmest person in the vicinity of a touchline shoving match is Diego Costa, but that was the situation Manchester City found themselves in at the end of their ill-tempered 3-1 home defeat by Chelsea. City dominated the game, took the lead and might have had three penalties, but Chelsea's clinical counter-attacking -- and a healthy dose of good fortune -- ensured it was Antonio Conte's side who prevailed, safeguarding their position at the league summit. City's frustrations told in stoppage time when Sergio Aguero's brainless lunge on David Luiz and Fernandinho's angry shove on Cesc Fabregas earned both players red cards that will keep them out for four and three games respectively, obliging the fiery Costa to assume the unfamiliar role of peacemaker. City manager Pep Guardiola apologised for his side's behaviour and most now make sure his players get the defeat -- and the nervous tension it engendered -- out of their systems as quickly as possible.

Liverpool tripped up by Achilles heel

If ever a game highlighted Liverpool's Achilles heel is their defence and endangers their title hopes this one will be it. Credit to substitute Ryan Fraser and his Bournemouth team-mates for the gutsy fightback but an absent left-back in James Milner twice allowed them to get behind the defence and led to goals. The positive is that even without Philippe Coutinho Liverpool are as entertaining and ruthless up front though Reds boss Jurgen Klopp may rue taking Sadio Mane off as early as he did after his team conceded three goals in the last 14 minutes of their stunning 4-3 defeat.

Sanchez swaggers for stylish Gunners

When Alexis Sanchez posed for a selfie with a group of ball-boys after his hat-trick in Arsenal?s 5-1 win at West Ham on Saturday, it was the closest anyone connected to the Hammers had managed to get to the Chile forward all evening. Sanchez was at his majestic best with a 14-minute treble in the second half, leaving West Ham boss Slaven Bilic to admit he felt humiliated. The former Barcelona star underlined his value to Arsenal by setting up Mesut Ozil's opener in opportunistic style, then scored in the 72nd, 80th and 86th minute, the last of the hat-trick a fittingly show-stopping number as he deftly clipped the ball over West Ham goalkeeper Darren Randolph.

Fellaini blunder adds to Mourinho woe

Just when Jose Mourinho must have thought Manchester United's luck was turning for the better, the Old Trafford chief was undone by his decision to turn to the error-prone Marouane Fellaini in Sunday's 1-1 draw at Everton. United had taken a first half lead through Zlatan Ibrahimovic when the Swede punished a mistake by Everton goalkeeper Maarten Stekelenburg and enjoyed more good fortune when Marcos Rojo escaped a red card for a terrible tackle. Yet United were still frustrated in the 89th minute when Fellaini, who had only just been sent on by Mourinho, gave away the penalty that Leighton Baines converted to rescue a point.

Leicester headed in wrong direction

A 'relegation six-pointer' for the champions this early in the season is worrying indeed and losing 2-1 to a Sunderland side that takes the ordinary out of extraordinary makes Leicester look like they have returned to the strugglers of old. The Foxes have made the worst start to a season of any Premier League champion with only 13 points from a possible 42 and are just two points above the drop zone. The problems run deep from Jamie Vardy's goal drought to Riyad Mahrez being totally out of sorts and new players such as Ahmed Musa taking a long time to settle.