Is there an endofunctor $U: \mathrm{Top} \to \mathrm{Top}$ (or from some good subcategory) such that $H_n(UX) = H_{n+1}(X)$ for any $n \geq 1$
2026-03-26 07:59:22.1774511962
Is there an analogue of the loop space for homology?
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I don't believe such a functor exists, unless you restrict your attention to contractible spaces, which is uninteristing.
The reason is that then you would have $H_n(X)=H_{n-1}(UX)=\ldots=H_{-1}(U^{n+1}X)=0$.