The Higson corona and ultrafilters

• 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)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ORTIZ-CASTILLO, YASSER F.; TOMITA, ARTUR HIDEYUKI; YAMAUCHI, TAKAMITSU. HIGSON COMPACTIFICATIONS OF WALLMAN TYPE. TSUKUBA JOURNAL OF MATHEMATICS, v. 42, n. 2, p. 233-250, DEC 2018. (14/16955-2, 16/23134-0, 17/04355-9, 16/26216-8)