Exercise 1, § 6

Let $K$ be one of the topological fields $\mathbf{R},\mathbf{C},\mathbf{H}$. Let $V$ be a vector space over $K$ possessing a countable generator system, endowed with its finest topology of topological vector space (EVT, I, §1, p. 11, cor. 4). For every integer $n\geqslant 1$, denote by $B_n(V)$ the subspace of $V^n$ formed by free sequences and by $G_n(V)$ the set of subspaces of dimension $n$ of V. Denote by $p: B_n(V)\rightarrow$ $G_n(V)$ the mapping which associates with a sequence $v\in B_n(V)$ the vector subspace which it generates and endow the set $G_n(V)$ with the least fine topology for which the mapping $p$ is continuous.

(a) Prove that one defines a right action of the group $\mathbf{G}\mathbf{L}(n,K)$ in $B_n(V)$ by putting $v\cdot g= (\sum^n_{i=1}v_ia_{i,j})_{1\leqslant j\leqslant n}$ for $v= (v_1, . . . , v_n)\in$ $B_n(V)$ and $g= (a_{i,j})\in \mathbf{G}\mathbf{L}(n,K)$.

(b) Prove that the mapping $p$ induces a homeomorphism of the quotient space $B_n(V)/\mathbf{G}\mathbf{L}(n,K)$ onto $G_n(V)$. Deduce that the space $G_n(V)$ is paracompact, and metrizable if V is of finite dimension.

(c) Prove that $B_n(V)$ is a principal fibre space with group $\mathbf{G}\mathbf{L}(n,K)$ and basis $G_n(V)$.

(d) Suppose that V is of infinite dimension. Prove that the space $B_n(V)$ is contractible. Deduce that the space $G_n(V)$ is a classifying space for the group $\mathbf{G}\mathbf{L}(n,K)$.