best, D

]]>Thanks for promoting the Taubes paper (and also for writing the other posts on this blog site). I look forward to investigating further!

Best wishes,

Paul.

]]>Once again many thanks!

Paul.

]]>I don’t think I’d ever claim anything like that.

]]>I note that at the end of Scorpan, the Fintushel-Stern conjecture states that K3(k’) and K3(k”) are diffeomorphic iff k’ and k” are equivalent knots (modulo reflections?) and that possibly all of the complexity of knot theory is contained in 4-manifold topology. Maybe this relates a little to the paper you recommend (?). I agree with you in that I like the style and broad sweep of the book. In it Scorpan also recommends a nice AT book called Algebraic Topology by Hatcher that clarifies a lot that I had previously found explained too cryptically in say the (original) book by Greenberg – and which is also much broader in scope than the latter.

Thanks for your recommendation of the Taubes 2016 paper and your perspective on this subject.

Paul.

]]>The most recent interesting paper on 4-manifolds that I read (well, skimmed) was http://arxiv.org/abs/1602.01687 by Cliff Taubes, where he gives examples of (families of) 4-manifolds for which certain families of anti-self dual metrics have “limits” of the form (S^3-K)xS^1 with a product metric, where K is a knot in the 3-sphere, and where S^3-K has its hyperbolic metric (so that the product is therefore conformally flat, and so trivially anti-self dual). This is a kind of connection between the 4 and 3 dimensional worlds that looks extremely important to me, but I have nothing nontrivial to add beyond that.

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