On the homotopy type of VMO (Q1380881)

Let \(X\) and \(Y\) be compact smooth manifolds without boundary. By \(\text{VMO} (X,Y)\) (vanishing mean oscillation) is understood the class of maps from \(X\) to \(Y\) introduced by \textit{H. Brezis} and \textit{L. Nirenberg} in their paper [Sele. Math., New Ser. 1, 197-263 (1995; Zbl 0852.58010)]. The set \(\text{VMO} (X,Y)\) is equipped with a metric paracompact topology and contains the space \(C(X,Y)\) as a proper dense subset. The author proves that the inclusion map \(i: C(X,Y)\hookrightarrow \text{VMO} (X,Y)\) is a homotopy equivalence. Combining this with a well-known theorem of Whitney asserting that the inclusion \(C^k(X,Y) \hookrightarrow C(X,Y)\) is a homotopy equivalence for \(1\leq k\leq\infty\), he obtains the, following sequence of homotopy equivalences: \[ C^\infty(X,Y) \hookrightarrow C^k(X,Y) \hookrightarrow\cdots C^1(X,Y) \hookrightarrow C(X,Y) \hookrightarrow \text{VMO}(X,Y). \] This implies that all the homotopy, homology and cohomology groups of \(\text{VMO} (X,Y)\) are isomorphic via the homomorphisms induced by inclusion to the corresponding groups of \(C^\infty(X,Y)\).