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Original post: https://math.stackexchange.com/questions/3810423/it-is-possible-that-x-simeq-%ce%a9x-and-that-x-simeq-%ce%a9-2x

I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 he proposes the following exercise

Let $ X $ be a path-connected non-contractible space.
(1) It is possible that $ X \simeq ΩX $ ?
(2) It is possible that $ X \simeq Ω^ 2X $ ?

My answers would be

(1) No, because otherwise $ \pi_n X \simeq \pi_{ n - 1 } \Omega X \simeq \pi_{ n - 1 } X \simeq \cdots \simeq \pi_0 X \simeq *$ and by Whitehead $X \simeq *$ (assuming $X$ CW)
(2) Yes, for example $X = \prod_{n=0}^\infty K (\mathbb{Z}, 2n+1)$ (or $X = U$ by Bott periodicity)

The problem is that at that chapter no Whitehead theorem nor Eilenberg-MacLane spaces (nor Bott periodicity) are available, so there would be a more down-to-earth answer!

What is available up to chapter 4?
Chapter 1: categories and functors.
Chapter 2: limits and colimits.
Chapter 3: convenient categories of spaces, i.e. with CW complexes, their limits and something more; smash product, smash-map adjuction, suspension, loospace.
Chapter 4: homotopies, contractible spaces, nullhomotopies, abstract homotopies with cylinder and path objects, (homotopy) groups and cogroups, homotopy groups; mapping spaces; maps over and under a space; CW structure on loospace.

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  • $\begingroup$ What is $U$ in question (2) ? $\endgroup$
    – Joël
    Commented Sep 3, 2020 at 16:13
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    $\begingroup$ @Joël Isn't that the underlying space of the infinite unitary group? I think Bott periodicity gives you deloopings Ω(BU×Z) ~ U and ΩU ~ BU×Z $\endgroup$
    – Harry Gindi
    Commented Sep 3, 2020 at 16:19
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    $\begingroup$ You can try asking the author himself: wmich.edu/math/directory/strom Anyway, I might get ahead of myself but maybe the book is simply not meant to be read 100% linearly. Maybe when you first read the chapter your answer should be "uh, I can't find any obstruction to that, weird" and when you read the rest you should think "ahah, now I know!". (Though I'd be happy to learn of an answer to the question as asked.) The question is "is it possible", not "find an example", after all. "With what I've learned so far, it's possible" is a fine answer... $\endgroup$
    – Najib Idrissi
    Commented Sep 3, 2020 at 17:03
  • $\begingroup$ @Joël The infinite unitary group $\endgroup$
    – CNS709
    Commented Sep 3, 2020 at 22:56
  • $\begingroup$ @NajibIdrissi Coulde be! I have seen that the author is a quite active user on MO, so maybe it is not necessary to contact him to get an answer! $\endgroup$
    – CNS709
    Commented Sep 3, 2020 at 22:58

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