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Tuesday Seminar on Topology
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2019/06/18
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masaki Taniguchi (The University of Tokyo)
Filtered instanton homology and the homology cobordism group (JAPANESE)
Masaki Taniguchi (The University of Tokyo)
Filtered instanton homology and the homology cobordism group (JAPANESE)
[ Abstract ]
We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.
We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.
