A filtration of the loops on SU\((N)\) by Schubert varieties
type: Item
instance of: scholarly article
title: A filtration of the loops on SU\((N)\) by Schubert varieties
zbMATH Open document ID: 0613.57025
DOI: 10.1007/BF01229802
author: Stephen A. Mitchell
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173790
review text: 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 \(G_{n,k}\).
Mathematics Subject Classification ID: 57T25
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 3992465
zbMATH Keywords: integral homology of \(\Omega \) SU\((n)\)
zbMATH Keywords: Thom spaces
zbMATH Keywords: Schubert varieties
zbMATH Keywords: loop groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3992465