Real Bott Tower

TL;DR

This paper classifies the diffeomorphism types of three- and four-dimensional real Bott towers, showing there are four and twelve distinct classes respectively, based on the action of $(bZ_2)^n$ on $(S^1)^n$.

Contribution

It establishes the exact number of diffeomorphism classes for 3D and 4D real Bott towers, providing a complete classification.

Findings

4 diffeomorphism classes in 3D

12 diffeomorphism classes in 4D

Classification based on group actions

Abstract

In this paper, we proved that there exist four distinct diffeomorphism classes of three-dimensional real Bott tower $M(A)=(S^1)^3/(\mathbb{Z}_2)^3$, and 12 distinct diffeomorphism classes of four-dimensional real Bott tower $M(A)=(S^1)^4/(\mathbb{Z}_2)^4$, where matrix $A$ corresponds to the action of $(\mathbb{Z}_2)^n$ on $(S^1)^n$ for n=3,4.