2007 Fiscal Year Final Research Report Summary
A Study on Controlled Surgery Theory
| Project/Area Number |
16540078
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Okayama University of Science (2005-2007)
Josai University (2004) |
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Principal Investigator |
YAMASAKI Masayuki Okayama University of Science, Fac. of Sci., Professor (70174646)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIZAWA Kiyoko Josai U., Fac. of Sci., Professor (90053686)
TSUCHIYA Susumu Josai U., Fac. of Sci., Associate Professor (60077914)
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| Project Period (FY) |
2004 – 2007
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| Keywords | surgery / assembly map / index |
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| Research Abstract |
1. We established a stability property of epsilon controlled L-groups for quite general control maps. The key ingredient was the local Alexander trick which shrinks things locally in certain directions. All the other known methods used splitting of quadratic Poincare complexes into small pieces, but this is not possible in the general setting. This stability result enabled us to understand other properties of epsilon controlled L-groups and identify the limit as the controlled surgery obstruction group. This is a joint work with E. Pedersen.
2. We proved several homology-like properties of the controlled L-groups, e.g. the MeyerVietoris sequence, the sequence for pairs, and excisions. We also observed the difference of the controlled L-groups and the homology. Using these we could give a new proof of the topological invariance of the rational Pontrjagin classes. This is a joint work with A. Ranicki.
3. Using the controlled surgery exact sequence of Pedersen-Quinn-Ranicki, we showed that the classical surgery sequence for certain 4-manifolds is exact at the normal structure set term.
4. The classical Poincare-Hopf theorem states that the signature of a vector field on a compact manifold X with only isokated singular points is equal to the Euler characteristic of X, if the vector field is never zero and points outward on the boundary. Morse extended this to the case where the vectors on the boundary are not assumed to be outward. We could extend this to the case where the singular points are allowed to be on the boundary but the induced vector filed on the boundary has only isolated singular points. This is a joint work with H. Kamae.
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Research Products
(28 results)
