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Aug 3, 2023 at 1:02 comment added Greg Friedman I know this is very late, but I've just been reading about this stuff, and according to a point in the thesis of Nussbaum, the homotopy groups alone do not determine such spaces as classifying spaces. So for posterity I wanted to leave a reference to it. It's Obstruction Theory of Possibly Nonorientable Fibrations by Frank Nussbaum, Northwestern 1970. See Comment 2.3.13. Here's a link for those with access: proquest.com/docview/…
May 24, 2019 at 13:51 comment added Patrick Elliott Thank you for the answer, the reference is well appreciated. Do you have any pointers on the second part of my question? What is obtained by such localisations? Precisely, what class of spaces is local in this sense?
May 20, 2019 at 7:57 history answered Mark Grant CC BY-SA 4.0