2007 Fiscal Year Final Research Report Summary
Infinitely generated objects
| Project/Area Number |
16540125
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Waseda University |
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Principal Investigator |
EDA Katsuya Waseda University, Faculty of Science and Engineering, Professor (90015826)
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| Project Period (FY) |
2004 – 2007
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| Keywords | wild space / fundamental group / homotopy group / infinite words / compact abelian group / grope group |
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| Research Abstract |
We have gotten results on the following five items. That is, we gotten considerably sufficient results for our original plans except for continuous words.(1) The fundamental groups of wild spaces ; (2) New constructions of wild spaces ; (3) A classification of finite-sheeted convering maps over 2-dimensional compact abelian groups ; (4) Grope groups ; (6) Infinitary words.
(1): Suppose that a Peano continuum X is wild everywhere, i.e. not locally semi-simply connected at every point and pi_1(X) is a subgroup of the free product G^*H. Then, pi_1(X) is a conjugate of a subgroup of G or H. This implies that the fundamental groups of wild Peano continua cannot only be taken part into free products, but have a completely opposite property. This result was announced at the occasion of Borsuk conference and was submitted in December of 2005. This result is applied to a study of the fundamental groups obtained by attaching copies of the Hawaiian aearring to manifolds.
(2) : In a collabolation with D. Repovs and U. Karimov we introduced a new construction of a space SC(X), which is obtained by attaching a cone C(X) to the square along the Topologists' sine curve. For a connected space SC(X) is simply-connected, and pi _2(4) is trivial if and only if pi_1(X) is trivial.
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