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The cohomological structure of fixed point set for pro-torus actions on compact spaces

In this paper, we study the relationships between the cohomological structure of a space and that of the fixed point set of a finite dimensional pro-torus action on the space. This paper is about finite dimensional compact groups. The concept of the dimension of a compact topological group plays an essential role in transformation group theory. | Turk J Math (2018) 42: 3164 – 3172 © TÜBİTAK doi:10.3906/mat-1806-55 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article The cohomological structure of fixed point set for pro-torus actions on compact spaces Mehmet ONAT∗, Department of Mathematics, Faculty of Science, Çukurova University, Adana, Turkey Received: 12.06.2018 • Accepted/Published Online: 15.10.2018 • Final Version: 27.11.2018 Abstract: In this paper, we study the relationships between the cohomological structure of a space and that of the fixed point set of a finite dimensional pro-torus action on the space. Key words: Totally nonhomologous to zero, pro-torus, fixed point 1. Introduction Most of the results in the cohomology theory of transformation groups (based on the Borel construction) concern Lie group actions and results about non-Lie group actions are still quite scarce. However, in the case of topological groups (without assuming smooth structure), other groups exist, e.g., p -adic integers or solenoid. These wild groups play an important role in, for example, Hilbert–Smith conjecture. The main difficulty in generalizing theorems for Lie group actions to topological group actions is how to handle these wild groups. In this paper, we try to overcome this difficulty by using the fact that for a finite dimensional compact group G , there is a closed normal subgroup N such that the quotient group G/N is a compact Lie group and by applying the cohomological technique of Lie group actions to show the main theorem. This paper is about finite dimensional compact groups. The concept of the dimension of a compact topological group plays an essential role in transformation group theory. The Lebesgue covering dimension or topological dimension of a compact Hausdorff space is defined as below. A collection A of subsets of the space X is said to have order n + 1 if some point of X lies in n + 1 members of A, and no point of X lies in more than n + 1 members of .