Filtered instanton Floer homology and the homology cobordism group

  • Yuta Nozaki

    Yokohama National University; Hiroshima University, Japan
  • Kouki Sato

    Meijo University, Nagoya, Japan
  • Masaki Taniguchi

    Kyoto University, Japan
Filtered instanton Floer homology and the homology cobordism group cover
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Abstract

For any and oriented homology 3-sphere , we introduce a homology cobordism invariant . The values are included in the critical values of the -Chern–Simons functional of , and we show a negative definite cobordism inequality and a connected sum formula for . 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 along a knot has the Frøyshov invariant negative, then all positive -surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use to define a filtration on the homology cobordism group which is parametrized by . Moreover, we compute an approximate value of for the hyperbolic 3-manifold obtained by -surgery along the mirror of the knot .

Cite this article

Yuta Nozaki, Kouki Sato, Masaki Taniguchi, Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. 26 (2024), no. 12, pp. 4699–4761

DOI 10.4171/JEMS/1371