Topology of the groups of upper triangular matrices

1. $U_n$. Let $\mathfrak{n}$ be the space of strictly upper triangular matrices, a vector space of dimension $n(n-1)/2$ (the free entries above the diagonal). Each $N\in\mathfrak{n}$ satisfies $N^n=0$ and $\exp(N)$ is upper triangular with ones on the diagonal, so $\exp(\mathfrak{n})\subset U_n$. Conversely $U\in U_n$ is $I+N$ with $N\in\mathfrak{n}$ and $\log(I+N)\in\mathfrak{n}$ (a polynomial in $N$ without constant term, so strictly upper triangular). By the previous problem $\exp:\mathfrak{n}\to U_n$ is a homeomorphism, so $U_n\cong\mathfrak{n}\cong\mathbb{R}^{n(n-1)/2}$.

2. $B_n$: a product decomposition. An invertible upper triangular $T$ has nonzero diagonal entries $d_i=t_{ii}$. Let $D=\mathrm{diag}(d_1,\dots,d_n)$. Then $D^{-1}T$ is upper triangular with ones on the diagonal, so $T=D\cdot U$ with $U=D^{-1}T\in U_n$. The map
$$B_n\to\mathcal{D}\times U_n,\qquad T\mapsto\bigl(D,\ D^{-1}T\bigr)$$
(where $\mathcal{D}\cong(\mathbb{R}^\times)^n$ is the group of invertible diagonal matrices) is continuous with continuous inverse $(D,U)\mapsto DU$. So $B_n\cong(\mathbb{R}^\times)^n\times U_n$ as topological spaces.

Components. $\mathbb{R}^\times=(-\infty,0)\cup(0,\infty)$ has two components, each homeomorphic to $\mathbb{R}$, hence contractible. So $(\mathbb{R}^\times)^n$ has $2^n$ components (choose a sign for each diagonal entry), each homeomorphic to $(0,\infty)^n\cong\mathbb{R}^n$. Multiplying by $U_n\cong\mathbb{R}^{n(n-1)/2}$ (connected and contractible) does not change the number of components, and each component of $B_n$ is homeomorphic to $\mathbb{R}^n\times\mathbb{R}^{n(n-1)/2}=\mathbb{R}^{n(n+1)/2}$, which is contractible.

Check. $n=1$: $B_1=\mathbb{R}^\times$, $2$ components. $n=2$: $B_2=\{\begin{pmatrix}a&b\\0&d\end{pmatrix}:ad\ne0\}$, $4$ components according to the signs of $a$ and $d$.

Remark. The signs of the diagonal entries are the only topological invariants of $B_n$: there are $2^n$ components, while $GL_n$ has only $2$ (connected by rotations, which are not triangular).