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I think the topological case is handled by this: Brown, Morton A mapping theorem for untriangulated manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 92–94 According to the review on MathSciNet, it shows that an $n$-manifold $M$ has the form $M = X \cup_\alpha D^m$ where $\dim(X) < n$. I don't ...


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For the smooth case: Via Morse theory the claim is equivalent to having a Morse function with only one maximum, or only one minimum, or to have a handle decomposition with only one 0-handle. Assume you have several 0-handles. By connectedness they have to be connected via 1-handles. Smale's handle cancellation says that a k-handle can be canceled against a ...


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This is not Taubes' moduli space. Taubes has many more constraints on the currents, so that his "moduli space" is really a finite set of points (for generic $J$), and he requires special weightings on the multiply-covered curves (which are necessarily unbranched covers of tori due to his constraints) to get a well-defined count from his set of currents. He ...



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