| Grant number: | 16/23134-0 |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |
| Start date: | August 28, 2017 |
| End date: | September 01, 2017 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
| Principal Investigator: | Artur Hideyuki Tomita |
| Grantee: | Artur Hideyuki Tomita |
| Visiting researcher: | Takamitsu Yamauchi |
| Visiting researcher institution: | Ehime University, Japan |
| Host Institution: | Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil |
Abstract
The Higson corona was introduced to study K-theory of C*-algebra but there are some studies in the direction of general topology (see for instance [3] and [6]). The topic that we are interested in is the covering dimension of Higson coronas, especially in Dranishnikov's problem (see [2] and [8]). In [7], Protasov used filters to deal with Higson coronas, and proved that, under CH, the Higson corona of a proper ultrametric space is homeomorphic to the remainder of the Stone-Cech compactification of w. A related study can be found in the paper of Banakh, Chervak and Zdomskyy [1]. It is unknown whether the Higson corona of the real line is homeomorphic to the remainder of the Stone-Cech compactification of the real line. They have similar properties, which can be found in, for example, [5] and [4]. We think that by using the properties of ultrafilters it is possible to find some partial answers to this problem. [1] T. Banakh, O. Chervak and L. Zdomskyy, On character of points in the Higson corona of a metric space, Comment. Math. Univ. Carolin. 54 2 (2013) 159-178. [2] A. Dranishnikov, Asymptotic Topology, Russian Math. Surveys 55, (2000) 1085-1129. [3] A. Dranishnikov, J. Keesling and V. V. Uspenskij, On the Higson corona of uniformly contractible spaces, Topology 37, (1998) 791-803.[4] Y. Iwamoto and K. Tomoyasu, Higson compactifications obtained by expanding and contracting the half-open interval, Tsukuba J. Math. 25 1 (2001), 179-186.[5] J. Keesling Subcontinua of the Higson corona, Topology Appl. 80 (1997), 1155-160.[6] J. Keesling, The one-dimensional Cech cohomology of the Higson compactification and its corona, Topology Proc. 19 (1994), 129-148[7] I. V. Protasov, Coronas of ultrametric spaces, Comment. Math. Univ. Carolin. 52 2 (2011) 303-307[8] T. Yamauchi, Hereditarily infinite-dimensional property for asymptotic dimension and graphs with large girth, preprint. (AU)
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