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Ricardo Andrade
  • Member for 12 years, 10 months
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From Topological to Smooth and Holomorphic Vector Bundles
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From Topological to Smooth and Holomorphic Vector Bundles
@Daniel: I did not know you are at Stanford as well. Perhaps I will look for you. :) You are indeed correct about my usage of the notion of smooth structure. I would say that the two smooth structures you describe on ${\mathbb R}$ are diffeomorphic but not the same/equivalent. That is sometimes a useful distinction, for example when studying spaces of smooth structures on manifolds, as one does in the theory of smoothings of topological or piecewise linear manifolds.
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From Topological to Smooth and Holomorphic Vector Bundles
@Daniel: I am glad the new wording reads better. I agree that the previous wording was rather excessive. Well, I must admit that really is not how I would interpret the question, given how it is stated. I certainly think that my example is rather silly, but it seems to me that it answers the question as written. Perhaps this all comes from my homotopical viewpoint, where I do not feel comfortable identifying two things just because they are isomorphic or homotopic, but must always specify how they are isomorphic or homotopic.
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From Topological to Smooth and Holomorphic Vector Bundles
@Michael: Are you asking about my definition of equivalence of two smooth structures on a topological manifold? I will assume so for this comment. For convenience, I will define a differentiable structure on a topological manifold $M$ to mean a maximal smooth atlas on $M$. Under that definition, I consider two differentiable structures on $M$ to be equivalent if the corresponding maximal smooth atlases are the same.
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From Topological to Smooth and Holomorphic Vector Bundles
(continuation) I absolutely agree that any two ways of endowing a topological vector bundle with a smooth structure which makes it into a smooth vector bundle give rise to isomorphic smooth vector bundles. However, as I attempt to argue in my answer above, that isomorphism cannot in general be the identity function on the total space. Perhaps I am misunderstanding what you mean. If so, please let me know.
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From Topological to Smooth and Holomorphic Vector Bundles
@Daniel: I understand you objection, and I apologize if my wording was too strong and unpleasant. I have changed it accordingly, and I hope it reads better now. Nevertheless, I will persevere with my statement, as it seems to me that there exist distinct, non-equivalent differentiable structures on the total space of a vector bundle which make it into a smooth vector bundle. I want to be quite clear that I consider a differentiable structure to be either a maximal smooth atlas, or an equivalence class of smooth atlases. (to be continued)
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On the (derived) dual to the James construction.
(continuation) This may be a simple observation others (including John Klein, perhaps) have realized on their own. Nevertheless, it may be a good idea to leave it as a comment here, in case someone else is able to make good use of it.
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On the (derived) dual to the James construction.
I was thinking about this question. I did not get very far, but it seems to me that $L(X)$ is the homotopy limit of the following semisimplicial space $F$ (i.e. $F$ is a functor from the opposite of the category of finite ordinals and order preserving monomorphisms to the category of spaces). On objects it is given by the wedge $F(T)=X^{\vee T}$. Seeing $X^{\vee T}$ as the space of functions $T\to X$ whose support has at most one point (as John Klein suggests), $F$ evaluated at a monomorphism of ordinals $f$ is simply pre-composition with $f$. (to be continued)
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Local finality condition (for re-indexing parameterized colimits)
@David: I am happy to hear that. You are most welcome. By the way, I corrected a few errors in my answer: some arrows were pointing in the wrong direction and had the wrong names. I hope there are no major issues remaining.
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Local finality condition (for re-indexing parameterized colimits)
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