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Sharafutdinov’s retraction for Alexandrov spaces.

The Sharafutdinov's retraction is defined for Open Manifolds of Nonnegative Curvature, for example in $\S 3.4$ of the book "Metric Foliations and Curvature" by Gromoll and Walschap.

For Alexandrov spaces, Perelman gave a constraction in $\S 6$ of his preprint "ALEXANDROV'S SPACES WITH CURVATURES BOUNDED FROM BELOW II". But the construction is not "that" rigous compare with the one given by Gromoll and Walschap. He definitly is a great mathematician, but I just want to make sure this is a correct construction. So are therey any published paper or more detailed construction available?

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 The same construction works; I do not see any difference. – Anton Petrunin Sep 30 at 22:53 Does the Sharafutdinov retraction is unique? Which is unique for smooth case, but the proof involves exponation map. – MG Oct 1 at 9:51 MG: you may find http:.//wwwmath.uni-muenster.de/u/awoer_01/files/diss_woerner.pdf useful. – Igor Belegradek Oct 1 at 14:33 MG: as Anton said, Sharafutdinov's construction carries over without change. Of course, this does not mean that Perelman's proof of the soul conjecture carries over. Before asking further questions, I suggest you glance through Woerner's thesis in the link above. – Igor Belegradek Oct 1 at 17:23