Framed instanton homology and concordance, II

We continue our study of the integer-valued knot invariants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu Superscript normal ♯ Baseline left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ν </mml:mi> <mml:mi mathvariant="normal"> ♯ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\nu ^\sharp (K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r 0 left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r_0(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We first establish a “conjugation” symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu Superscript normal ♯ Baseline left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ν </mml:mi> <mml:mi mathvariant="normal"> ♯ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\nu ^\sharp (K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon Superscript normal ♯ Baseline left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ϵ </mml:mi> <mml:mi mathvariant="normal"> ♯ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\epsilon ^\sharp (K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Hom’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi> ϵ </mml:mi> <mml:annotation encoding="application/x-tex">\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .