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Abstract:

A 3-manifold is said to be SU(2)-abelian if all homomorphisms from its fundamental group to SU(2) factor through the first homology of the manifold. Understanding which manifolds are SU(2)-abelian is a difficult and wide open problem, even in the case of 3-manifolds arising from surgery on a knot in the 3-sphere. Kronheimer and Mrowka showed, for instance, that n-surgery on a nontrivial knot is not SU(2)-abelian for n = 1 or 2, but it isn’t even known whether the same is true for n = 3 or 4 (the same isn’t true for n = 5 as 5-surgery on the right-handed trefoil is a lens space, which has abelian fundamental group). We approach this question by trying to understand which knots have instanton L-space surgeries. A rational homology sphere is said to be an instanton L-space if its framed instanton homology has the smallest rank possible, in analogy with Heegaard Floer homology; familiar examples include lens spaces and branched double covers of alternating knots. Moreover, it is generally the case that SU(2)-abelian manifolds are instanton L-spaces. We conjecture that if surgery on a knot K in the 3-sphere results in an instanton L-space then K is fibered and its Seifert genus equals its smooth slice genus, and we discuss a strategy for proving this.