Scalable spaces

Degrees of maps and multiscale geometry

“…Similarly, 3 , and so finally, we have deg 𝑓 𝐿 3 . If we have a weaker low-frequency assumption that 𝑃 ≤ l 𝑎 𝑖 = 𝑎 𝑖 for every i, then the same argument shows that deg 𝑓 2 l 𝐿 3 .…”

“…It is not difficult to construct an L -Lipschitz self-map of with degree . When or , then [3] shows that there are also L -Lipschitz self-maps of with degree . But when , [3] shows that every L -Lipschitz self-map of has degree .…”

“…Next, we will get a contradiction by considering the wedge product.Let 𝑊 : Λ 2 R 4 × Λ 2 R 4 → Λ 4 R 4 be the quadratic form given by the wedge product. It has signature(3,3). Now, let 𝐵 ⊂ Λ 2 R 4 be the subspace spanned by 𝑏 1 , .…”

Lipschitz homotopies of mappings from 3‐sphere to 2‐sphere

“…In [1], this process of coarsening of a mapping (and hence, building $$\sim L$$‐Lipschitz homotopies) was generalized to a broad class of spaces $$Y$$, dubbed scalabe . Name comes from the defining feature of having self‐maps $$Y\rightarrow Y$$ that scale $$Y$$ well, in the sense that they act on $$H^n(Y,\mathbb {R})$$ by $$L^n$$ while having Lipschitz constant $$\sim L$$.…”

“…And in the limit these forms give a presentation of cohomology $$H^\bullet (Y)\rightarrow \Omega _\flat (Y)$$ (morally, because $$f*d$$ is scaled out). And vice versa, such presentation is a guide for shadowing principle to construct scaling mappings — and hence, efficient homotopies: Theorem For a (formal simply connected finite) complex $$Y$$ the existence of $$\sim L_t\times L_x$$ null‐homotopies in $$[X,Y]$$ for any (finite) $$X$$ is equivalent to the existence of a presentation $$H^\bullet (Y)\rightarrow \Omega _\flat (Y)$$ of DGA of its cohomology (sending each class to its representative) [1]. …”