Nik, I was not realy aware of "accept" mark on answers.I did not have intension to not show courtesy to people who answer my question.I thank you for inform me of this "accept" mark. but just a question: could not you send me a personal message(email) for this subject? thanks again for your comment

your interesting question is a motivation for the following related question: is the equivariant version of conformal classification of topological annuli, true?(see Riemann Surface,Farkas and Kra) Namely, assume that V is an open subset of the plane,with annuluar topology, which is invariant under the action of the isometry group of the plane. Does there exist an equivariant biholomorphic map between V and some annular reagin in the plane? If the answer to this question is affirmative, then your question has also an affirmative answer.

Mike thank you for your beutiful answer. But do you mean that an arbitrary symplectomorphism is necessarily equivariant with respect to O(n) action?

Tim,Daniel,Mike and Francois, thanks a lot for your helps

Igor thank you very much for your answer

@Qiaochu If f:X ---->Y is a homotopy equivalent then f*(pull back) gives a natural bijection between vect_{n}(X) and Vect_{n}(Y), the isomorphism class of n dimensional bundles on X and Y, respectively

Yes in my question I mean symplectomorphism. but how lema 6.11 implies that they are sympletctomrphism?

@Qiaochu a restriction is existence of a vector bundle which is not a summand of a trivial bundle Ex;the canonical line bundle on RP^\infty, see vector bundles and k theory by Allen Hatcher

What type of vector bundles we should associate to a lsc multimap on X ? Under my assumption can we prove a weaker result that each lsc multimap on X with valuse in finite dimensional banach space possesses a continuous selection?

Dear Julius Korbas, in the Stong paper, is there a restriction on k and n, when he state his result oabout G(k,n)? If there is no restriction, it seems that some thing is missing. Because a similar argument as above proof of fixed point property of even dimensional projective space, can be repeated to give a proof for fixed point property for odd dimensional projective space. But it is well known that CP^n has no FPP when n is odd