Kasparov's immortal game vs Topalov is often regarded as one of the best games in chess history.
However, after analyzing it, I find that objectively speaking, Kasparov played some bad moves and got into a slightly worse position. Here, in this position below, Kasparov sacrificed his rook on d4 and Topalov made a mistake by capturing it.
[White "Garry Kasparov"]
[Black "Veselin Topalov"]
[FEN "b2r3r/k4p1p/p2q1np1/NppP4/3p1Q2/P4PPB/1PP4P/1K1RR3 w - - 0 24"]
1. Rxd4 cxd4 2. Re7+ Kb6 3. Qxd4+ Kxa5 4. b4+ Ka4 5. Qc3 Qxd5 6. Ra7 Bb7
7. Rxb7 Qc4 8. Qxf6 Kxa3 9. Qxa6+ Kxb4 10. c3+ Kxc3 11. Qa1+ Kd2 12. Qb2+ Kd1
13. Bf1 Rd2 14. Rd7 Rxd7 15. Bxc4 bxc4 16. Qxh8 Rd3 17. Qa8 c3 18. Qa4+ Ke1
19. f4 f5 20. Kc1 Rd2 21. Qa7 1-0
However, if Topalov had played 24...Kb6! then Kasparov's initiative would have come to nothing.
[White "Garry Kasparov"]
[Black "Veselin Topalov"]
[FEN "b2r3r/k4p1p/p2q1np1/NppP4/3p1Q2/P4PPB/1PP4P/1K1RR3 w - - 0 24"]
1. Rxd4 Kb6! 2. b4 Qxf4 3. Rxf4 Nxd5 4. Rxf7 cxb4 5. axb4 Nxb4 6. Nb3 Bd5 7. Rf6+
Nc6 8. f4 Rhf8 9. Rxf8 Rxf8
and Black is slightly better.
If we compare this position to the position after move 16, White does seem slightly better.
[White "Garry Kasparov"]
[Black "Veselin Topalov"]
[FEN "r3k2r/1b1nqp1p/p1pp1npQ/1p2p3/3PP3/P1N2P2/1PP1N1PP/1K1R1B1R w kq - 0 13"]
The question therefore is, where did Kasparov misplay the position? How did he lose his small positional advantage. Where is the improvement for White? Or is White's positional advantage an illusion? Is the position objectively equal or even better for Black?

g4will give you anything as Black will castle long... Best regards. – AlwaysLearningNewStuff Jul 26 '14 at 20:09unclearin his comments ( see my answer ). Going backward is ridiculous, as you are going towards the starting opening moves. If an improvement was possible in move 12 or before, the entire opening would be refuted, which is not the case. Please see Kasparov's entire commentary about this game from Chess Informant 74. As I have said before: White is "pretty" but is slow. Black is "ugly" but ideally placed and hass all pieces developed... – AlwaysLearningNewStuff Jul 26 '14 at 21:40zipfile from the edited part of my answer. BTW, you were right,...Bg7was imprecise but as I have said, Kasparov's game was even worse in the beginning which he explained in his comments. Also take a look at Seirawan's analysis. This is the best I can do... Good luck, you are on your own Spartan... – AlwaysLearningNewStuff Jul 29 '14 at 22:36