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I happen upon this old thread now. Maybe it is still worth giving an update, and more of an answer. The latest account (as of the time of this writing) of the conjectural statement in question here appears as Conjecture 1.17 in Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology in H. Sati, U. Schreiber (eds.) ...


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You have many comments to the effect that "topology is needed to describe continuity, calculus concepts, the notion of "looks like", homeomorphism and so forth". And these are all altogether right, but I'm getting that your question is about the global picture. Also, the following is mainly about a toplological or differentiable manifold; Joshphysics's link ...


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Strictly speaking, if there can be defined charts covering a set (an atlas), you can give that set the topology induced by defining the charts to be bicontinous. That is, a set is open iff it's the domain of a chart in the maximal atlas. If your set already has a topology, the topology induced by the atlas will agree with that one under some conditions. (I ...


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First let me refer you to Eric Weinberg's book where the instanton moduli space is described in more detail. Principal bundles over 4-dimensional Riemannian manifolds are classified by the second Chern class = Instanton number and the t' Hooft discrete Abelian magnetic fluxes. Please see the following Lecture notes by Måns Henningson. t' Hooft fluxes ...



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