Description

• Title:
  Almost flat bundles and almost flat structures
• Creator:
  Mishchenko, Alexander S.;
  Teleman, Nicolae
• Description:
  In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [< i> Conjecture de Novikov et fibrés presque plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (1990), 273–277]. Using a natural construction of [B. Hanke and T. Schick, < i> Enlargeability and index theory< /i> , preprint, 2004], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles over arbitrary CW-spaces using quasi-connections [N. Teleman, < i> Distance function, Linear quasi-connections and Chern character< /i> , IHES/M/04/27].< /p>< p> Connes, Gromov and Moscovici [< i> Conjecture de Novikov et fibrés presque plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (1990), 273–277] showed that for any almost flat bundle $\alpha$ over the manifold $M$, the index of the signature operator with values in $\alpha$ is a homotopy equivalence invariant of $M$. From here it follows that a certain integer multiple $n$ of the bundle $\alpha$ comes from the classifying space $B\pi_{1}(M)$. The geometric arguments discussed in this paper allow us to show that the bundle $\alpha$ itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace $Y\subset B\pi_{1}(M)$ trough the classifying mapping.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2005.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 26, No 1 (September 2005); 75 - 87;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.025/9278
• Subject:
  Almost flat bundles;
  quasi-connection;
  classifying space;
  higher signature;
  19L41;
  19K56;
  19J25;
  55N15
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.025
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other