Homotopic maps in the general linear group

Prove that the maps $\mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\longrightarrow \mathrm{GL}(2n,\mathbb R)$ that send $(A,B)$ to the block matrices $\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$ and $\begin{pmatrix} AB & 0\\ 0 & 1\end{pmatrix}$ are homotopic.

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One thought on “Homotopic maps in the general linear group”

1. The solution we came up with Iván:

The homotopy is given by $(A,B,t)\mapsto \begin{pmatrix} A & 0\\ 0 & B\end{pmatrix} \begin{pmatrix} 1 & tB\\ 0 & 1\end{pmatrix} \begin{pmatrix} 1 & 0\\ -tB^{-1} & 1\end{pmatrix}\begin{pmatrix} 1 & tB\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1 & -t\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1 & 0\\ t & 1\end{pmatrix}\begin{pmatrix} 1 & -t\\ 0 & 1\end{pmatrix}$, and it is in fact smooth. This actually proves that the subspace of matrices of the form $\begin{pmatrix} B & 0\\ 0 & B^{-1}\end{pmatrix}$ is contractible in $\mathrm{GL}(2n, \Bbb{R})$.

Some other minor observation: this is some kind of homotopical version of Whitehead’s lemma in K-theory.

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