A filtration of the loops on SU\((N)\) by Schubert varieties
The integral homology of \(\Omega SU(n)\) is known to be a polynomial algebra on classes \(b_ 1,...,b_{n-1}\) with dim \(b_ i=2i\). This gives a filtration of \(H_*(\Omega SU(n))\) by the subgroups \(R_{n,k}\) of polynomials of degree at most k in the \(b_ i\). This paper proves the conjecture of Mahowald and Hopkins, establishing the following theorem: There is a filtration \(F_{n,k}\) of the space \(\Omega\) SU(n) which realizes the filtration \(R_{n,k}\) in homology. Further, there is a commutative diagram \[ \begin{tikzcd}\Omega \mathrm{SU}(n) \rar & BU \\ F_{n,k} \rar \uar & BU(k)\uar \end{tikzcd} \] so that the successive quotients in the filtration are the Thom spaces of the induced vector bundles. The reference to Schubert varieties in the title is an analogy for the theory of ''loop groups'' of the classical Schubert cell decomposition in the theory of algebraic groups. The theory of loop groups is used explicitly to construct the spaces \(G_{n,k}\).
integral homology of \(\Omega \) SU\((n)\)
scientific article; zbMATH DE number 3992465
scientific article