2019 Fiscal Year Annual Research Report
Understanding exotic spheres from the viewpoint of global singularity theory of smooth maps
| Project/Area Number |
18F18752
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| Research Institution | Kyushu University |
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Principal Investigator |
佐伯 修 九州大学, マス・フォア・インダストリ研究所, 教授 (30201510)
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| Co-Investigator(Kenkyū-buntansha) |
WRAZIDLO DOMINIK 九州大学, マス・フォア・インダストリ研究所, 外国人特別研究員
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| Project Period (FY) |
2018-11-09 – 2021-03-31
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| Keywords | bordism of smooth maps / B_2 singularity / fold singularity / smooth map germ / homotopy sphere / SKK-group / signature / intersection space |
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| Outline of Annual Research Achievements |
We have computed the n-dimensional bordism group of Morse functions on compact manifolds possibly with boundary except for n=4k+1. Our result generalizes previous works of Ikegami, Saeki, and T. Yamamoto. Our approach uses explicit methods of geometric topology that can handle B_2 singularities at boundary points. As an application, we obtained new topological invariants for generic smooth map germs at boundary points into the plane. As for our study of the bordism group of Morse functions with index constraints, we have related this group to a certain SKK-group by means of a natural map. We study the properties of this map in ongoing work by using handle exchanges from surgery theory. Moreover, we have investigated a short exact sequence that contains the bordism group of Morse functions with index constraints. Namely, we have shown that a certain candidate for a splitting is not available in dimensions of the form n=4k+3. Our progress in these topics strengthens the impact of global singularity theory on homotopy spheres. In another part of our research, we have obtained new results about intersection spaces, which are a spatial construction due to Banagl that gives access to Poincare duality and the signature of singular spaces. Namely, in ongoing work, we are using differential forms to construct a nondegenerate intersection pairing for the intersection spaces of a class of singular spaces including toric varieties. Moreover, in joint work with Essig, we have constructed a fundamental class for intersection spaces of certain singular spaces of stratification depth two.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We are preparing a paper about our study of the bordism group of Morse functions on compact manifolds possibly with boundary. We have presented our results on a poster at a conference in Tokyo, and in talks at conferences in Kiew and in Rio de Janeiro. In the oriented setting, the case of dimension n=4k+1 remains to be understood (including some resulting invariants of smooth map germs). For Morse functions on closed manifolds, one has an additional bordism invariant based on the Kervaire semi-characteristic in these dimensions, but it is not clear whether an analogous invariant arises in our setting or not. Our approach to the study of bordism groups of Morse functions on compact manifolds was originally inspired by the observation that they are naturally related to SKK-groups. We are preparing a paper that discusses important consequences of this connection. For manifolds with boundary, we introduce and study relative SKK-groups with the goal to compute the admissible fold bordism group of Morse functions. We also relate the bordism group of Morse functions with index constraints to SKK-groups of highly connected manifolds, and explain the impact of our results to concrete homotopy spheres. We have submitted a paper about nondegenerate intersection pairings for the intersection spaces of singular spaces with isolated singularities. Currently, we are preparing a paper that generalizes this toy model to singular spaces with trivial link bundles along the intersection space construction of Agustin-Bobadilla (2018). Our joint paper with Essig will be submitted in May 2020.
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| Strategy for Future Research Activity |
Apart from completing our papers in preparation, we plan to continue ongoing research in the following topics. We continue to study our groups of standard special generic maps that provide a singularity theoretic filtration of the group of homotopy spheres. In doing so, we hope to advance the challenging computation of the Gromoll filtration. Our long-term goal is to develop in arbitrary dimension a systematic obstruction theory for elements of our filtration to lie in the Gromoll filtration. Even in dimension 7, the Gromoll filtration is not entirely computed. For this reason, we are interested in the construction and existence of standard special generic maps on the classical homotopy 7-spheres constructed by Milnor. Another direction of research is bordism theory of fold maps into higher dimensional target spaces. This relates to recent work of Kalmar who computed the structure of bordism groups of fold maps into the plane. From our perspective, fold maps into higher dimensional target spaces can be used to define a singularity theoretic double filtration of the group of homotopy spheres. This double filtration unifies the bordism group of Morse functions with index constraints with our group of standard special generic maps. We are interested in the position of concrete homotopy spheres in this double filtration. As basis for our methods we use helpful geometric constructions including plumbing, fiber bundles, and Brieskorn spheres. The third aspect of our ongoing research takes first steps towards high-dimensional signature formulas in the spirit of Saeki-Yamamoto.
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