Filtered instanton Floer homology and the homology cobordism group

For any s E [-cc , 0] and oriented homology 3-sphere Y, we introduce a homology cobordism invariant rs(Y ) s (Y ) E (0, cc ]. The values { r s (Y ) } are included in the critical values of the SU (2)-Chern-Simons functional of Y, and we show a negative definite cobordism inequality and a connected sum formula for rs. s . As applications, we obtain several new results on the homology cobordism group. First, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. Next, we show that if the 1-surgery of S3 3 along a knot has the Fr & oslash;yshov invariant negative, then all positive 1=n-surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use {rs} r s } to define a filtration on the homology cobordism group which is parametrized by [0, cc ]. Moreover, we compute an approximate value of rs s for the hyperbolic 3-manifold obtained by 1=2-surgery along the mirror of the knot 52. 2 .