Calculations of characteristic numbers of immersed manifolds by REDUCE

2001 Fiscal Year Final Research Report Summary

• Project/Area Number: 11640055
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Hirosaki University
• Principal Investigator: KIKUCHI Shigeki Hirosaki University, Faculty of Science and Technology, Department of Mathematical Science, Assistant Professor, 理工学部, 助教授 (30003510)
• Project Period (FY): 1999 – 2000
• Keywords: manifold / immersion / characteristic numbers / cobordism class

Research Abstract

In 1983, R. Stong studied cobordizm classes of compact smooth manifolds immersed in euclidean space with codimension three. Their cobordism classes are undetermined in cases of their dimensions are of some special types. The first unknown case occurs in dimension 16.
Therefore we investigated characteristic numbers of immersed manifolds with codimension three by using REDUCE. We out putted Wu classes of such manifolds and knew all Stiefel- Whitney numbers were zero. This means that the manifold is a boundary. Programming are not difficult, so that we will be able to apply them in another dimension. The version of REDUCE is 3.2, it is very old today. We used it on IBM Intellistation Pentium III model with a MO drive. In the case of dimension is 16 it worked well. But as numbers of partitions of a given dimension n increase very rapidly, memories of PC will be overflow. Thus we will face another problem to handle our hardware well. It is a computer problem different from so called pure mathematics, we hope many proceedings to it. Nowadays to determine unknown problems in pure mathematics by using computer is going to be a big current. We would like to follow up undetermined cobordism problems as above.