Real secondary index theory

Bunke, Ulrich; Schick, Thomas

doi:10.2140/agt.2008.8.1093

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Real secondary index theory

Ulrich Bunke and Thomas Schick

Algebraic & Geometric Topology 8 (2008) 1093–1139

DOI: 10.2140/agt.2008.8.1093

Abstract

In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to what extent the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by $8$ ) which can be defined in this case contains extra information over the complex index (the index of its complexification). We study this question under the additional assumption that the complex index vanishes on the $k$ –skeleton of $B$ . In this case, we define new analytical invariants $ĉ_{k} \in H^{k - 1} (B; ℝ ∕ ℤ)$ , certain secondary invariants.

We give interesting nontrivial examples. We then describe this invariant in terms of known topological characteristic classes.

Keywords

family index, secondary characteristic classes

Mathematical Subject Classification 2000

Primary: 57R20

References

Publication

Received: 12 July 2005

Revised: 23 May 2008

Accepted: 27 May 2008

Published: 6 July 2008

Authors

Ulrich Bunke
Universität Regensburg Mathematische Fakultät 93040 Regensburg Germany
http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke
Thomas Schick
Georg-August-Universität Göttingen Mathematisches Institut Bunsenstr. 3 37073 Göttingen Germany
http://www.uni-math.gwdg.de/schick

Volume 8, issue 2 (2008)