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I have briefly fixed the clause for topological spaces at contractible space, making manifest the distinction between contractible and weakly contractible.
Is it worth an example of a weakly contractible but not contractible space, such as the double comb space?
I don’t have time for it, but if you do, it would sure be worth it.
Ok. I rearranged things as the same thing was being said in two different places.
Thanks!
I have hyperlinked cohesive ∞-groupoid.
Added examples:
Examples
A(n inhabited) convex subset of a topological vector space over ℝ (or ℂ) is contractible.
The unit sphere in a separable infinite-dimensional Hilbert space is contractible, unlike the case of finite-dimensional spheres
By Kuiper’s theorem the unitary group of such a Hilbert space is contractible, where the topology can be either of the two main topologies (norm topology, or strong operator topology).
The total space of any universal principal bundle is contractible.
Have hyperlinked more terms, such as: inhabited convex subset, infinite-dimensional sphere, separable Hilbert space , finite-dimensional sphere, unitary group U(ℋ), norm topology, strong operator topology.
Ta. I was in an airport and pressed for time before boarding.
In fact Illusie proved in
that the group of bounded operators of any Hilbert space is (weakly) homotopy equivalent to a point. I’m not sure if Palais’ theorem that is used for the separable Hilbert space case that gets actual contractibility applies here.
In any case Illusie seems to be making the stronger claim about contractibility outright. And also has the statement about U(H). All in the norm topology (this reference is noted at Kuiper’s theorem, I now see, though Wikipedia claims that Kuiper only stated his theorem for the case of separable H, and Illusie’s paper shows it for arbitrary H)
Also this should be added (when I have time) as a citation for a strengthening of the current claim about unit spheres in Hilbert spaces:
I see infinite-dimensional sphere only makes the weaker statement that the unit sphere in a normed space is weakly contractible. The article in the previous comment should go there too, when I have time.
Re #12: I’m confused. Off the bat, the only group structure I see on the space of bounded operators B(H) is the addition structure, and there contractibility is obvious. Did you mean to say that or something else like “invertible” replacing “bounded”?
Sorry! It was very late at the time, and I may have gotten the wrong idea from the French… Re-reading the slightly non-trivial sentence (with subordinate clauses and so on) in the theorem in the light of day, I see he means the topological group of invertible operators with the norm topology (inherited from the space bounded operators).
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