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In the paper "Orbispaces, orthogonal spaces, and the universal compact Lie group" by Stefan Schwede, he studies (spaces with an action of) the topological monoid $\mathbf{L}(\mathbb R^\infty,\mathbb R^\infty)$ of linear isometric self-embeddings of ${\mathbb R}^\infty$ equipped with the subset topology of $\operatorname{maps}(\mathbb R^\infty,\mathbb R^\infty)$ with compact-open-topology. The underlying space of this monoid is contractible (e.g. Remark A.12). What is known about the homotopy type of the subgroup of invertible elements of this monoid, i.e. the linear isometric self-isomorphisms of ${\mathbb R}^\infty$? In particular, is the underlying space still contractible?

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There are two related questions for which the answer is known. Put $H_0=\mathbb{R}^\infty$, and let $H$ be the Hilbert space completion of $H_0$. Let $G_0$ be the colimit of the orthogonal groups $O(\mathbb{R}^n)$. Let $G_1$ be the group of pairs $(g,h)\in\mathbf{L}(H_0,H_0)^2$ with $fg=gf=1$, topologised as a subspace of a product. Let $G_2$ be the group of invertible elements of $\mathbf{L}(H_0,H_0)$. Let $G_3$ be the group of all orthogonal automorphisms of $H$. We then have continuous injective homomorphisms $$ G_0 \xrightarrow{i_0} G_1 \xrightarrow{i_1} G_2 \xrightarrow{i_2} G_3 $$ The space $G_0$ is not contractible; it has $8$-periodic homotopy groups starting $$ \mathbb{Z}/2,\mathbb{Z}/2,0,\mathbb{Z},0,0,0,\mathbb{Z}. $$ You asked about the space $G_2$. The map $i_1$ is a continuous bijection but I suspect that it is not a homeomorphism. Equivalently, I suspect that the inversion map on $G_2$ is not continuous, so $G_2$ is not a topological group. So you might find that it is better to think about $G_1$. In any case, I do not think that the homotopy type of $G_1$ or $G_2$ is known. However, it is known that $G_3$ is contractible. I am not aware of any natural applications of $G_1$ or $G_2$, although they certainly seem to be natural objects. You might consider whether it would be better for you to use $G_0$ or $G_3$.

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  • $\begingroup$ Thank you really much for your answer! I was not aware of the problem that $G_2$ might not be a topological group (I also assumed this for some applications I had in mind). I want to work with $\mathcal L$-spaces as a model for orbispaces as suggested in Schwede's paper. The space $G_2$ came up when I tried to understand homeomorphisms between $\mathcal L$-spaces or embeddings of (proper Lie) groupoids into $\mathcal L$. I don't think that I can use $G_0$ or $G_3$ without changing the whole model. Maybe it's best to avoid this issue of invertible elements of $\mathcal L$. $\endgroup$
    – Branko J
    Oct 18 '19 at 11:13

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