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Something seems missing. Should the third line include the condition $deg(f) = 0$? Because we can’t have both $n\; odd, deg(f) \neq 0 \Rightarrow n\; odd \Rightarrow Maps_f(S^n, S^n) \simeq_\mathbb{Q} S^n$ and $n\; odd, deg(f) \neq 0 \Rightarrow Maps_f(S^n, S^n) \simeq_\mathbb{Q} S^{2n-1}$.
Thanks for catching this. The first “odd” should have been the second “even”. Fixed now.
Thanks.
added pointer to
based on
added statement of the rational cohomology of $\Omega^D S^n$, for $1 \leq D \lt n = 2k$ (here)
am adding this statement also to iterated loop space
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