Carter only managed a 1-over 71 in the third round, while Fisher shot a 2-under 68, one better than Alker's 69 on Saturday. The trio finished 54 holes at 9-under-par 201.
David Higgins (68), Thomas Levet (68), Phillip Price (67) and Joost Luiten (71) are knotted in fourth place at minus-7 at Kennemer Golf and Country Club.
Carter struggled with bogey at the par-5 second hole, but answered with a birdie at the fifth. He hit a poor third shot at the seventh and missed a 28- foot par save. Carter lost another stroke at the par-3 eighth when his tee ball came up short and his second did not even reach the putting surface.
Carter was 8 under par for the championship and did not collect another birdie until the par-5 12th when his drive left him with a wedge for his second. He two-putted from 30 feet, but came up short with his approach at 13 en route to a bogey.
His fortunes changed at the par-3 15th. Carter nearly holed his tee ball for an ace, but settled for a tap-in birdie to join the lead at minus-9. At the next hole, Carter drove into the right rough, but managed to get his approach on to the left fringe. He sank the 12-footer from off the green for a birdie and sole possession of first at 10 under par.
Carter made a clutch 6-footer to save par at 17, then drove into the left rough at the closing hole. He could not get to the green with his second, but unfortunately, left his third short of the putting surface as well. Carter putted to a foot and tapped in for the bogey that cost him the lead.
Fisher flew out of the gate with birdies at one, four and five, with the last two coming from inside 5 feet. He birdied the eighth, but bogeyed three in a row from the ninth, including a pair of missed 6-footers.
Fisher birdied the 12th to tie for the lead at 9 under, but a bogey at the 13th dropped him down the leaderboard. He gave himself a good look at birdie from 12 feet at the 15th, but missed.
At the 16th, Fisher ran home a 7-foot birdie putt to once again tie for the lead. He had a putt of similar length at 17, but came up short. Fisher's 25-foot birdie try for the lead at the last grazed the left edge.
Alker was a par machine with nine in a row to start. He rolled in a 4-foot birdie putt at the 10th and reached 10-under par for the tournament with a two-putt birdie at the par-5 12th.
He yanked a 3-foot par putt at 13 for his first bogey since his seventh hole in the opening round. Alker continued his par plan with three in a row, but appeared to be in trouble at the par-3 17th.
Alker's tee ball got hung up in the wind and fell into a right greenside bunker. His blast from the trap did not reach the putting surface, but no matter. Alker chipped in for par, then two-putted from 25 feet for a par at the last.
John Bickerton (69), Stephen Gallacher (66), Alexander Noren (72) and 1999 British Open champion Paul Lawrie (69) are tied for eighth place at 6-under 204.
Three-time winner Bernhard Langer, who will make his Champions Tour debut next week, fired a 5-under 67 and is part of a group tied for 12th at minus-5.