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@ThomasRot what about the smoothneds of tge resulting surjective map. Approximation in $C^r$ topology $r>0$ is at hand but what about approximation of continuous object with smooth one?chapter 2 of Hirsch book
@Z.M namely if a manifold is a countable union of open sets $U_n \subset U_{n+1}$ where $U_n$ is homeomorphic to $R^k$ then the manifold is homeomorphic to $R^k$
In this example I think the plane $A$ is the pull back of the real tangent bundle of $\mathbb{C}P^2$ under the fiber map $S^5 \to \mathbb{C}P^2$. The tangent bundle of CP^2 is irreducible but it apparently does not implies that the bundle A you mentioned is irreducible. So this is a motivation to ask: What is an example of principle bundle $X\to B$ where $A$ is a connection plane tangent to X such that $TB$ is irreducible but A is not? I would appreciate if you give comment on this question
@ThomasRot BTW you told the torus cover every manifold what is your argument for a least possible degenerate map? Because I remember a *** exercise of Hirsch book which says: Every manifold a countable union of homeomorphic copy of open disk is R^n. So I guess that the mapping you would provide is very degenerate
@ThomasRot Yes I agree that dimX=dim Y implies that we have a finite covering space.But I meant arbitrary dimension. In this case we have a classificaton of fibre bundle.
@ThomasRot Any way I think that analytic objects are very interesting.As you know cartan solved the local embedding theorem in case of analytic case. So they are worth of attention. For example I wonder the if torus covering you mentioned work in analytic case?
@ThomasRot your sentence in the Holomorphic setting does not work unless the base space would be Kahler. In the real case it is too degenerate. But what about the extra non degeneracy condition, say submersion, what are some examples that base space is not torus: $\mathbb{T}^n \to M$ M is not tori?
@ThomasRot Analytic objects are discussed in Hirsch Differential topology(For upgrading smooth structures to analytic one) and also Kobayashi and Nomizu Differential geometry. Some times passing from smooth to analytic would creat interesting advantures. I know splitting principal in the CW complex setting so please ellaborate your comment in smooth case. I would appreciate if you provide an answer.
@abx Please read my last question again. The injectivity of $f^*$ is no longer assumed. moreover do you mean $T_X$ instead of $T_Y$? Note that $Y$ is parallelizable. So I do not get your comment.
However the tangent bundle of $S^5$ is not decompossible in the way you mentioned but according to the Splitting principal there is an space $X$ and continuous map $f:X\to S^5$ such that $f^*$ is an isomorphism in K theory and the pull back $f^*(TS^5)$ of the tangent bundle of $S^5$ is decomposed to direct sum of line bundle. In this case s there an explicit (compact) manifold $X$ with this property?What is theminimum possible dimension for such manifold $X$? By this question I search for a possible smooth version of splitting principle in K theory
