Real Bott Tower

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