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Jul 8, 2023 at 7:16 comment added Yeah Yes I am asking for more examples than those I have mentioned. I am looking for examples so that the topology of $LX$ is simple and the homology groups are explicitly computable. Maybe they could not be as simple as the negatively curved case, but for example, could there be non-positively curved manifolds (other than flat tori) such that topology of the free loop space is also very simple?
Jul 7, 2023 at 6:49 comment added Achim Krause What type of description are you looking for? It seems to me that you've already determined the homotopy type completely in terms of the fundamental group: it's $K(C_{\alpha}, 1)$, where $C_{\alpha} $ is the centralizer. Are you asking for examples of aspherical manifolds where these centralizers are interesting and we have geometric descriptions for their classifying spaces?
Jul 7, 2023 at 5:51 history edited Yeah CC BY-SA 4.0
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Jul 7, 2023 at 5:26 history edited Yeah CC BY-SA 4.0
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Jul 7, 2023 at 4:36 history asked Yeah CC BY-SA 4.0