2018 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 | cobordism of smooth maps / Morse theory / surgery theory / fold map / elimination of cusps / signature / homotopy sphere / SKK-group |
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| Outline of Annual Research Achievements |
Our major achievement is the establishment of an explicit new relation between so-called SKK-groups of manifolds on the one hand and cobordism groups of Morse functions on the other hand. More precisely, we have constructed an explicit isomorphism between the oriented fold cobordism group of Morse functions on closed manifolds and the oriented SKK-group of closed manifolds. An interesting observation is that the signature of manifolds is invoked in the construction of the isomorphism. This is in accordance with our project goal to relate the signature, an algebraic key invariant of manifolds, to singularity theoretic phenomena. As a byproduct of our approach via SKK-groups we are able to compute unoriented and oriented cusp cobordism groups of Morse functions on compact manifolds with boundary of dimension at least 2. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. We also show that our notion of cusp cobordism yields the same cobordism groups as the notion of admissible cobordism previously studied by Saeki and Yamamoto who have solved the case of Morse functions on unoriented compact surfaces with boundary by a different approach. Therefore, our result answers open problems posed by Saeki and Yamamoto in 2015, and yields an explicit description of their cobordism invariant which they constructed by means of the universal complex of singular fibers. For non-singular Morse functions we show that the induced cobordism class is trivial, and recover the Morse-van Schaack equality on odd dimensional manifolds.
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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
One of our research goals is to study open questions concerning cobordism groups of Morse functions in the presence of Morse index constraints. Motivation comes from the fact that exotic Kervaire spheres can be distinguished from other exotic spheres as elements of such cobordism groups in infinitely many dimensions. In our attempt to study these cobordism groups, we were naturally led to our approach via SKK-groups. This approach provides a fruitful framework that allows to study many types of cobordism groups of Morse functions on manifolds from a unified perspective including cobordism of constrained Morse functions. We have presented the main ideas of our approach at a topology research meeting at RIMS in December 2018, and we prepare an unrefereed contribution to the proceedings. The power of our approach is visible from the fact that we are led to the solution of open problems posed by Saeki and Yamamoto concerning the cobordism groups of Morse functions on compact manifolds with boundary. The resulting paper will be submitted to a refereed journal in April 2019. Due to this unforeseen application of our approach we have decided to study in detail its ramifications before returning in FY2019 to the original plan to study cobordism groups of Morse functions in the presence of Morse index constraints by using our approach via SKK-groups which is now available. As for our plan to study singularity theoretic filtrations of the group of homotopy spheres, we have obtained valuable feedback from discussions in the context of our talk in the Oberseminar at Munster University.
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| Strategy for Future Research Activity |
Our recent approach via SKK-groups yields a unified framework that opens the way to study many types of cobordism groups of Morse functions on manifolds from the same viewpoint. We plan to exploit this approach by following our goal to study cobordism groups of Morse functions with index constraints. The idea is to express these cobordism groups in terms of certain SKK groups, which can then be studied independently of singularity theory. Another future direction can be to extend our approach to cobordism theory for fold maps into higher dimensional target spaces. This will be related to recent work of Kalmar who computed the structure of cobordism groups of fold maps into the plane. In view of our framework, it then becomes obvious to ask for a substitute of SKK-groups that should arise in the case of higher target dimensions. The generalization to higher target dimensions is also motivated by our vision to unify our singularity theoretic filtrations of the group of homotopy spheres to a double filtration. This double filtration will include as special cases the cobordism group of Morse functions with index constraints and our group of standard special generic maps, which is closely related to the Gromoll filtration. Our long-term goal is to develop a systematic obstruction theory for elements of our filtration to lie in the Gromoll filtration. The third part of the project takes first steps towards high-dimensional signature formulas. As a first step, we plan to extend Gromov's planar Morse inequality to fold maps of bordisms with non-empty boundary into
Euclidean spaces.
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